# This is the current version of the main file. # Comment: for merely showing existence of 5-part of III(J/Q), # maybe it will be enough to do the local computation on # the easier J(Q)/phitilde(Jtilde(Q)). Suppose I have found the # rank precisely (for example rank 0) with a 2-descent, # then I just have to identify an element explicitly in # the image of all the J(Q_p)/phitilde(Jtilde(Q_p)), to prove # that it is in the Selmer group. # Check whether -1,2,5, etc, are 5th powers in each other's Q_p. # Note: a similar principle might be applicable to just do Ed's # example in one direction, to get 5-part of Sha (even if not the # full Selmer group) for his case. # This file will eventually set up the Kummer map from: # y^2 = (5*r/4)*(r^2-3*r*x+3*x^2)^2 + (x-r)^5 # = (r/4)*(r^2-5*r*x+5*x^2)^2 + x^5 # The following are of order 5 on the Jacobian: # T1 = [ (r,G(r)) - infty ] = [ (r,(sqrt(5*r)/2)*r^2) - infty] # T2 = [ (0,H(0)) - infty ] = [ (0,(sqrt(r)/2)*r^2) - infty] # where: G(x) = (sqrt(5*r)/2)*(r^2-3*r*x+3*x^2), # and: H(x) = (sqrt(r)/2)*(r^2-5*r*x+5*x^2). # # We might as well just use the case r=4: # y^2 = 5*(16 - 12*x + 3*x^2)^2 + (x - 4)^5 # = (16 - 20*x + 5*x^2)^2 + x^5 # The following are of order 5 on the Jacobian: # T1 = [ (4,G(4)) - infty ] = [ (4, 16*sqrt(5)) - infty] # T2 = [ (0,H(0)) - infty ] = [ (0, 16) - infty] # Note that r can be reinserted via a quadratic twist: # (x,y) |--> (r*x, sqrt(r)*r^2*y). # Note let B_{ij} be the usual biquadratic forms: # ( B_{ij}(k(D_1),k(D_2)) ) # = ( k_i(D_1 + D_2)*k_j(D_1 - D_2) + k_i(D_1 + D_2)*k_j(D_1 - D_2) ). # # Let R_{ij} = B_{ij}( k(D), k(T_1) ) # = ( k_i(D + T_1)*k_j(D - T_1) + k_i(D - T_1)*k_j(D + T_1) ) # Let Rd_{ij} = B_{ij}( k(D), k(2*T_1) ), # = ( k_i(D + 2*T_1)*k_j(D - 2*T_1) + k_i(D - 2*T_1)*k_j(D + 2*T_1) ) # Similarly: # Let S_{ij} = B_{ij}( k(D), k(T_2) ) # = ( k_i(D + T_2)*k_j(D - T_2) + k_i(D - T_2)*k_j(D + T_2) ) # Let Sd_{ij} = B_{ij}( k(D), k(2*T_2) ), # = ( k_i(D + 2*T_2)*k_j(D - 2*T_2) + k_i(D - 2*T_2)*k_j(D + 2*T_2) ) # Then the sum of the orbit of: # k_i(D)*k_j(D+T_1)*k_k(D-T_1)*k_l(D+2*T_1)*k_m(D-2*T_1) # over all permutations of i,j,k,l,m gives 5! = 120 terms, # and this sum is unchanged by: D |--> D + T_1. # Note that; # k_i*R_{jk}*Rd_{lm} # gives 4 of these terms, and can take: # k_i*R_{jk}*Rd_{lm} + k_i*R_{jl}*Rd_{km} + k_i*R_{jm}*Rd_{kl} # + k_i*R_{kl}*Rd_{jm} + k_i*R_{km}*Rd_{jl} + k_i*R_{lm}*Rd_{jk} # to give 24 of the terms. The sum as you replace i with i,j,k,l gives # the entire expression: # k_i*R_{jk}*Rd_{lm} + k_i*R_{jl}*Rd_{km} + k_i*R_{jm}*Rd_{kl} # + k_i*R_{kl}*Rd_{jm} + k_i*R_{km}*Rd_{jl} + k_i*R_{lm}*Rd_{jk} # + k_j*R_{ik}*Rd_{lm} + k_j*R_{il}*Rd_{km} + k_j*R_{im}*Rd_{kl} # + k_j*R_{kl}*Rd_{im} + k_j*R_{km}*Rd_{il} + k_j*R_{lm}*Rd_{ik} # + k_k*R_{ji}*Rd_{lm} + k_k*R_{jl}*Rd_{im} + k_k*R_{jm}*Rd_{il} # + k_k*R_{il}*Rd_{jm} + k_k*R_{im}*Rd_{jl} + k_k*R_{lm}*Rd_{ji} # k_l*R_{jk}*Rd_{im} + k_l*R_{ji}*Rd_{km} + k_l*R_{jm}*Rd_{ki} # + k_l*R_{ki}*Rd_{jm} + k_l*R_{km}*Rd_{ji} + k_l*R_{im}*Rd_{jk} # k_m*R_{jk}*Rd_{li} + k_m*R_{jl}*Rd_{ki} + k_m*R_{ji}*Rd_{kl} # + k_m*R_{kl}*Rd_{ji} + k_m*R_{ki}*Rd_{jl} + k_m*R_{li}*Rd_{jk} # # Similarly for S,Sd. # This should give an enormous system invariant under D |--> D + T_1 # and another invariant under D |--> D + T_2. By tracking coeffs # of k4^5, ..., I should be able to pick out 8 indep for each, # and check that all others are dependent on these, and then # find the intersection, hopefully of dimension 4, to give the map. # Note: when checking LI, take into account the eqn of the Kummer, # so maybe replace all occcurrences of k2^2*k4^2 with 4*k1*k2*k4^2, # so that LI is then transparent. # I should be able to find a nice version of the map by making # the initial terms something like: k4^5, k4^4*k3, k4^4*k2, k4^4*k1. # # The rest of this file is currently as for pn2 (which I shall change). ######################################################################### # This file sets up the map between the Kummer surfaces. # # ######################################################################### interface(prettyprint,prettyprint=false): ######################################################################### # Let C be the curve: # C: Y**2 = f6*X**6 + f5*X**5 + f4*X**4 + f3*X**3 + f2*X**2 + f1*X + f0. # # For any divisor represented by {(x,y),(u,v)}, # let (k1,k2,k3,k4) be the embedding of the Kummer surface given by: # k1 := 1: k2 := x+u: k3 := x*u: k4 := (f0xu-2*y*v)/((x-u)**2): # ## where f0xu is: # f0xu := 2*f0+f1*(x+u)+2*f2*(x*u)+f3*(x+u)*(x*u) # +2*f4*(x*u)**2+f5*(x+u)*(x*u)**2+2*f6*(x*u)**3: # # Then the defining equation of the Kummer surface is given by # the following homogeneous quartic: kummeqn. ######################################################################### # R,S,T are as follows: kumR := k2**2-4*k1*k3: kumS := -4*k1**3*f0-2*k1**2*k2*f1-4*k1**2*k3*f2-2*k1*k2*k3*f3 -4*k1*k3**2*f4-2*k2* k3**2*f5-4*k3**3*f6: kumT := -4*k1**4*f0*f2+k1**4*f1**2-4*k1**3*k2*f0*f3 -2*k1**3*k3*f1*f3-4*k1**2*k2**2 *f0*f4+4*k1**2*k2*k3*f0*f5-4*k1**2*k2*k3*f1*f4-4*k1**2*k3**2*f0*f6+2*k1**2*k3** 2*f1*f5-4*k1**2*k3**2*f2*f4+k1**2*k3**2*f3**2-4*k1*k2**3*f0*f5+8*k1*k2**2*k3*f0 *f6-4*k1*k2**2*k3*f1*f5+4*k1*k2*k3**2*f1*f6-4*k1*k2*k3**2*f2*f5-2*k1*k3**3*f3* f5-4*k2**4*f0*f6-4*k2**3*k3*f1*f6-4*k2**2*k3**2*f2*f6-4*k2*k3**3*f3*f6-4*k3**4* f4*f6+k3**4*f5**2: kummeqngeneral := kumR*k4**2+kumS*k4+kumT: # For any divisor represented by A = {(x,y),(u,v)}, # let (k1,k2,k3,k4) be the embedding of the Kummer surface given by: # k1 := 1: k2 := x+u: k3 := x*u: k4 := (f0xu-2*y*v)/((x-u)**2): # ## where f0xu is: # f0xu := 2*f0+f1*(x+u)+2*f2*(x*u)+f3*(x+u)*(x*u) # +2*f4*(x*u)**2+f5*(x+u)*(x*u)**2+2*f6*(x*u)**3: # Similarly let the divisor B = {(x',y'),(u',v')} have the # Kummer coordinates: (l1,l2,l3,l4). # # Then the following 4 x 4 array BBB is such that, projectively: # ( BBB[i,j] ) = ( k_i(A+B) k_j(A-b) + k_i(A-B) k_j(A+B) ) with(linalg): BBB := matrix(4,4): BBB[1,1] := k1^2*l4^2+k4^2*l1^2+k3^2*l2^2*f5^2+l3^2*k2^2*f5^2-4*l4*k3^2*f6*l1+2*k3*l2*l4*k1 *f5+4*k3*l2*l4*f6*k2-2*l4*k1*k4*l1-4*l3^2*k1*f3*k2*f6+4*l3*k1*l2*f3*k3*f6+8*l3* k1*f2*k3*f6*l1+4*l3*l4*k1*f6*k3+2*l3*k4*k2*f5*l1-4*k1^2*l3^2*f6*f2-4*l2^2*k1^2* f6*f0-2*l3*k3*f5^2*k2*l2+8*k3*f6*k2*l3*l2*f4-4*l3^2*k1*f6*k4-4*k4*k3*l2^2*f6+4* l3*f1*k1*f6*k2*l1+4*l3*k4*k2*l2*f6+4*k3*f6*k2*l3*f3*l1+4*l3*k4*k3*f6*l1-4*k2^2* f6*l3^2*f4-2*k4*k3*l2*f5*l1+4*f1*l2*f6*k1*k3*l1-4*k3^2*f6*l2^2*f4-4*k3^2*f6*l2* f3*l1-4*k3*f6*k2*f1*l1^2-4*k3^2*f6*f2*l1^2-2*l3*l4*k1*f5*k2-4*l3*l4*k2^2*f6-4* k1^2*l3*l2*f6*f1+8*l2*f0*f6*k1*k2*l1-4*k2^2*f6*f0*l1^2: BBB[1,2] := k4^2*l2*l1+k1*k2*l4^2+l3*l4*k1^2*f3-l4*k1*f3*k3*l1-l2*l4*k1*k4-l2*l4*k3*f5*k2+ l3*l4*f5*k2^2-l3*k2*l2*f5*k4-k1*k4*l3*f3*l1+l3^2*k1*f3*k2*f5-k3*f5*k2*l3*f3*l1+ k3^2*l2*f5*f3*l1-l3*k1*l2*f3*k3*f5+k4*k3*l2^2*f5+k4*k3*f3*l1^2-l4*k4*k2*l1+2*l3 ^2*k2^2*f6*f3-2*l2*l4*k1*f4*k3+2*l3*l4*k1*k2*f4-4*l3*f1*f6*k1*k3*l1+2*l3^2*f1* k1^2*f6-4*l3*k3*f6*k2*f2*l1+2*l3^2*k4*k1*f5+2*l3^2*k4*k2*f6-2*k1*k2*l3*f5*f1*l1 -4*l3*f0*f6*k1*k2*l1-4*l3*k1*f2*k3*f5*l1+2*l3*f1*k1*f6*k2*l2-2*l3*f1*f6*k2^2*l1 -2*l3*k4*k3*l2*f6-4*l3*k3*l2*f6*k2*f3-2*l3*k4*k2*f4*l1-2*l3*k4*k3*f5*l1+4*l3^2* f6*k2*k1*f2-2*l3*l4*k3*f6*k2+2*l3^2*k1^2*f5*f2-4*l3*k1*f2*k3*l2*f6+4*l3*l2*f0* k1^2*f6+2*l3*l2*f1*k1^2*f5-2*f1*l2^2*f6*k1*k3+2*k3^2*l2^2*f6*f3+2*k4*k3*l2*f4* l1+4*l2*f2*k3^2*f6*l1+2*l2^2*f0*f5*k1^2+2*f1*f5*k3*k2*l1^2-2*f1*l2*f5*k1*k3*l1+ 2*f1*l2*f6*k3*k2*l1-4*l2*f0*f5*k1*k2*l1+4*f0*k3*f6*k2*l1^2+2*k3^2*f5*f2*l1^2+2* f1*k3^2*f6*l1^2+2*l2*l4*k3^2*f6+2*l4*k3^2*f5*l1-2*l3*l4*k3*k1*f5-4*l2*f0*k1*f6* k3*l1+2*f0*f5*k2^2*l1^2: BBB[1,3] := k1*k3*l4^2+l3*k4^2*l1-l3^2*k3^2*f5^2-f1^2*k1^2*l1^2+l4*f1*k1*k2*l1-l2*l4*k4*k2+ l3*k3*f5*k4*l2-k1*k3*l3*f3^2*l1+l3*l4*k3*f5*k2+k1*k3*l2^2*f5*f1+l3*k2^2*f5*f1* l1+l2*f1*k1*k4*l1+l3*l4*k1*k4+2*l3*l4*k3^2*f6+l2*l4*k1*f3*k3+l4*k4*k3*l1+2*l4* k1^2*f0*l1+k1*k3*f3*l3^2*f5-2*l3^2*f1*k1*f6*k2+2*l3^2*k4*k3*f6+4*l3^2*k3^2*f6* f4+l3*k1^2*f3*f1*l1+l3*k3^2*f5*f3*l1+l3*k4*k2*f3*l1+2*l3^2*k3*f6*k2*f3-2*k1^2* f0*l2*l3*f5+4*l2^2*f0*f6*k2^2+2*l2^2*f0*f5*k1*k2+2*k3*l2^2*f6*k2*f1-l3^2*f1*k1^ 2*f5+l3*k2*l2*f5*k1*f1+2*l3*k2^2*l2*f6*f1+2*l3*k3^2*l2*f6*f3+2*f0*k1*k4*l1^2+2* l2*f0*f5*k2^2*l1-2*f0*f5*k3*k2*l1^2+2*f0*f3*k1*k2*l1^2+k1*f3*k3*f1*l1^2-2*l2*f1 *k3^2*f6*l1+2*k1^2*f0*l2*f3*l1+k3*l2*f5*k2*f1*l1+4*f0*k1^2*f2*l1^2-k3^2*f5*f1* l1^2+2*l4*f2*k1*k3*l1+4*k1*k3*l3*f6*f0*l1+2*l3*l4*k1*f4*k3+4*l3*k2*l2*f6*k3*f2+ 2*k1*k3*l3*l2*f5*f2+4*k1*k3*l3*f4*f2*l1+2*l3*k1*f4*k2*f1*l1-4*l3*l2*f0*k1*f6*k2 +2*l3*k1*f2*k4*l1+2*l3*k2*f5*k3*f2*l1+2*l3*k4*k3*f4*l1+4*l2*f0*k1*k2*f4*l1-4*l2 *f0*k3*f6*k2*l1+2*f1*l2*f4*k1*k3*l1: BBB[1,4] := k1*k4*l4^2+l4*k4^2*l1-l2*k4*k2*f5*f1*l1-k3*l2*f5*k1*f1*f3*l1-l3*l4*k1*f3*k3*f5+ k1*k4*f1*f3*l1^2-l3*l4*k2*f5*k4-l2*l4*f1*k1*k2*f5+l3^2*k2*f5^2*k3*f3+l3*k2^2*l2 *f5^2*f1-l2*l4*k4*k3*f5+l4*k1^2*f3*f1*l1-f1*k1*k2*f5*l3*f3*l1+l3*k3^2*f5^2*l2* f3-k4*k3*f5*l3*f3*l1+k3*l2^2*f5^2*k2*f1+2*l3*l4*k3^2*f5^2+2*f0*f5^2*k1^2*l3^2-4 *l2*f1*f6*k1^2*f0*l1+2*l2*f5*k1^2*f1^2*l1+2*l3^2*k3*f5^2*k4-4*l2*f0*f5*f6*k3^2* l1-8*l2^2*f0*k1*f6*k4+4*f0*f6*k1*k3*l2*f3*l1-4*l2*f0*f6*k2*k4*l1-4*f1*k3*f6*k1* l2*f2*l1-6*l2*f0*f5*k1*k4*l1-2*k4*k2*l2^2*f6*f1-2*k3*l2^2*f6*k1*f1*f3-2*l2^2*f0 *f5^2*k1*k3-4*k3*f6*k2*f1*l2*f3*l1-4*l2*f0*f5*f4*k1*k3*l1-8*f0*f6*k2^2*l2*f3*l1 -4*l2*f0*f5*f3*k1*k2*l1-2*l2*f0*f5^2*k3*k2*l1+2*l2*f6*k1*k2*f1^2*l1-8*l2*f0*f5* k1^2*f2*l1-16*l2*f0*f6*k1*k2*f2*l1+2*l2^2*f6*k1^2*f1^2-2*f1*l2*f6*k3*k4*l1-8*f0 *f5*k1*k2*f2*l1^2+2*f1^2*k1*k2*f5*l1^2-4*f0*k1*f6*k2*f1*l1^2+2*k1^2*f4*f1^2*l1^ 2-8*f0*f6*k2^2*f2*l1^2+8*f0*f2*f6*k1*k3*l1^2-4*k3*f6*k2*f3*f0*l1^2+2*k3^2*f6*f3 *f1*l1^2+2*k2^2*f6*f1^2*l1^2-8*k3^2*f6*f4*f0*l1^2-4*f0*k3*f6*k4*l1^2-4*f1^2*f6* k1*k3*l1^2-4*f0*f3*f5*k1*k3*l1^2-8*f0*f4*k1^2*f2*l1^2+2*f0*f3^2*k1^2*l1^2-8*f6* k1^2*f0^2*l1^2-4*f0*f4*k1*k4*l1^2-2*f0*f5*k2*k4*l1^2+2*k3^2*f5^2*f0*l1^2-4*l2* l4*f0*k1*f6*k2-4*l2*l4*k3*f6*k2*f2-4*l3*l4*k1*f2*k3*f6-2*l3*l4*f1*k1*f6*k2-6*l3 *l4*k4*k3*f6+2*l4*k1*f2*k4*l1-8*l3*l4*k3^2*f6*f4-4*l4*f4*k1^2*f0*l1-4*l4*k3*f6* k2*f1*l1-8*l4*f0*f6*k2^2*l1-2*l4*k3*k1*f5*f1*l1-4*l3*k3*l2*f5*k2*f6*f1-8*k3*f6* k2*f2*l3*l2*f4-2*l2*l4*k1*f2*k3*f5-2*l2*l4*k3^2*f6*f3-6*l4*f0*k1*k2*f5*l1-4*l3* k1*l2*f3*k2*f6*f1-2*l2*l4*f0*f5*k1^2+4*l4*f0*f6*k1*k3*l1-4*l3*l4*k3*f6*k2*f3-2* l2*l4*k2^2*f6*f1-4*l3*l4*f6*k1^2*f0-4*l3^2*f0*f5*k1*f6*k2-8*k3^2*f6*f4^2*l3^2-4 *k3*f6*k2*f3*l3^2*f4+4*l3^2*k3^2*f5*f6*f3-8*l3^2*k3*f6^2*k2*f1+2*k3^2*f5^2*l3^2 *f4-8*f0*f6^2*k2^2*l3^2+2*l3^2*k1^2*f3*f6*f1-8*f6*k1^2*f0*l3^2*f4-4*k2^2*f6*f1* l3*l2*f4-4*l3*l2*f0*f5*f6*k2^2-4*l3*l2*f0*k1^2*f6*f3-2*l3*l2*f0*f5^2*k1*k2+8*l3 *k1*f3*k3*f1*f6*l1-2*l3^2*k4*k2*f6*f3-2*k1*k4*l3*f5*f1*l1-2*l3*k4*k2*f5*f2*l1-8 *l3^2*k3*f6*k4*f4+8*l3^2*f0*k1*f6^2*k3-4*l3^2*k3*f5*k1*f6*f1-2*l3^2*k1*f3^2*k3* f6-8*l3^2*f2*k3^2*f6^2+4*l3*f0*f6*f3*k1*k2*l1-8*l3*f0*f4*f6*k1*k3*l1+4*l3*f0*f5 ^2*k1*k3*l1-2*l3*f0*f5^2*k2^2*l1-8*l3*f2^2*k3*f6*k1*l1-4*l3*f1*k3^2*f6*f5*l1-4* l3*k3*f5*k1*f4*f1*l1+4*k1*k4*l3*f6*f0*l1+8*l3*f0*f6^2*k3^2*l1-2*k3^2*f6*f3^2*l3 *l1-4*l3*f0*f4*k1*k2*f5*l1-2*k2^2*f6*f1*l3*f3*l1-4*k3*f6*k2*f2*l3*f3*l1-2*l3*k3 *f5^2*k2*f1*l1-4*l3*k1*f2*k2*f1*f6*l1+8*l3*f2*f6*k1^2*f0*l1-4*l3*f0*f5*k1^2*f3* l1-4*l3*k1*f2*k3*l2*f6*f3-2*l3*k3*l2*f5^2*k1*f1-4*l3*k4*k3*f2*f6*l1-4*k3^2*f6* f3*l3*l2*f4-8*l3*l2*f1*f6^2*k3^2+2*l3*k3*f5^2*k2*l2*f2-4*k1*k4*l3*l2*f6*f1-4*l3 *k4*k2*l2*f6*f2-4*l3*k4*k3*l2*f6*f3-4*l3*f1^2*k1^2*f6*l1-8*f0*f6*k2^2*l2^2*f4-8 *l2^2*f2*f6*k1^2*f0-8*l3*l2*f0*f6^2*k3*k2-8*l2^2*f0*f6*f3*k1*k2+2*l2^2*f0*f5^2* k2^2-4*k3*f6*k2*f1*l2^2*f4-8*l2^2*f0*f6^2*k3^2-4*l2^2*f0*f6*k3*k2*f5: BBB[2,1] := k4^2*l2*l1+k1*k2*l4^2+l3*l4*k1^2*f3-l4*k1*f3*k3*l1-l2*l4*k1*k4-l2*l4*k3*f5*k2+ l3*l4*f5*k2^2-l3*k2*l2*f5*k4-k1*k4*l3*f3*l1+l3^2*k1*f3*k2*f5-k3*f5*k2*l3*f3*l1+ k3^2*l2*f5*f3*l1-l3*k1*l2*f3*k3*f5+k4*k3*l2^2*f5+k4*k3*f3*l1^2-l4*k4*k2*l1+2*l3 ^2*k2^2*f6*f3-2*l2*l4*k1*f4*k3+2*l3*l4*k1*k2*f4-4*l3*f1*f6*k1*k3*l1+2*l3^2*f1* k1^2*f6-4*l3*k3*f6*k2*f2*l1+2*l3^2*k4*k1*f5+2*l3^2*k4*k2*f6-2*k1*k2*l3*f5*f1*l1 -4*l3*f0*f6*k1*k2*l1-4*l3*k1*f2*k3*f5*l1+2*l3*f1*k1*f6*k2*l2-2*l3*f1*f6*k2^2*l1 -2*l3*k4*k3*l2*f6-4*l3*k3*l2*f6*k2*f3-2*l3*k4*k2*f4*l1-2*l3*k4*k3*f5*l1+4*l3^2* f6*k2*k1*f2-2*l3*l4*k3*f6*k2+2*l3^2*k1^2*f5*f2-4*l3*k1*f2*k3*l2*f6+4*l3*l2*f0* k1^2*f6+2*l3*l2*f1*k1^2*f5-2*f1*l2^2*f6*k1*k3+2*k3^2*l2^2*f6*f3+2*k4*k3*l2*f4* l1+4*l2*f2*k3^2*f6*l1+2*l2^2*f0*f5*k1^2+2*f1*f5*k3*k2*l1^2-2*f1*l2*f5*k1*k3*l1+ 2*f1*l2*f6*k3*k2*l1-4*l2*f0*f5*k1*k2*l1+4*f0*k3*f6*k2*l1^2+2*k3^2*f5*f2*l1^2+2* f1*k3^2*f6*l1^2+2*l2*l4*k3^2*f6+2*l4*k3^2*f5*l1-2*l3*l4*k3*k1*f5-4*l2*f0*k1*f6* k3*l1+2*f0*f5*k2^2*l1^2: BBB[2,2] := k4^2*l2^2+k2^2*l4^2+2*l3^2*k3^2*f5^2+2*f1^2*k1^2*l1^2-4*l3*l4*k1*k4-4*l3*l4*k3^ 2*f6+2*l2*l4*k1*f3*k3-4*l4*k4*k3*l1-4*l4*k1^2*f0*l1-2*k1*k3*f3*l3^2*f5-4*l3^2* f1*k1*f6*k2-4*l3^2*k4*k3*f6-8*l3^2*k3^2*f6*f4-2*l3*k1^2*f3*f1*l1-2*l3*k3^2*f5* f3*l1+2*l3*k4*k2*f3*l1-4*l3^2*k3*f6*k2*f3-4*k1^2*f0*l2*l3*f5-8*l2^2*f0*f6*k2^2- 4*l2^2*f0*f5*k1*k2-4*k3*l2^2*f6*k2*f1-2*l3^2*f1*k1^2*f5-4*l3*k2*l2*f5*k1*f1-4* l3*k2^2*l2*f6*f1-4*l3*k3^2*l2*f6*f3-4*f0*k1*k4*l1^2-4*l2*f0*f5*k2^2*l1-4*f0*f5* k3*k2*l1^2-4*f0*f3*k1*k2*l1^2-2*k1*f3*k3*f1*l1^2-4*l2*f1*k3^2*f6*l1-4*k1^2*f0* l2*f3*l1-4*k3*l2*f5*k2*f1*l1-8*f0*k1^2*f2*l1^2-2*k3^2*f5*f1*l1^2+l3^2*k1^2*f3^2 +k3^2*f3^2*l1^2-4*k2^2*l3^2*f6*f2-2*l3*l4*k2*k1*f3-2*l2*l4*f5*k3^2-2*l2*l4*f1* k1^2-2*l4*f3*k3*k2*l1-4*l3^2*f2*f5*k2*k1-4*l3*l4*k1^2*f2-4*f1*k1^2*l3*l2*f4+8* l3*f0*k2^2*f6*l1+8*l3*k3*f6*k2*f1*l1+8*l3*f1*f5*k1*k3*l1+8*l3*f0*f5*k2*k1*l1-4* k4*k1*l3^2*f4-4*l4*f4*k3^2*l1+8*l2^2*f0*f6*k1*k3-4*l2^2*f2*f6*k3^2-4*k1^2*f2*l3 ^2*f4-4*k1^2*f0*l3^2*f6+8*l3*l2*f1*k1*f6*k3-2*l3*l2*f3*k1*k4-2*l3^2*f5*k2*k4-4* k1^2*f0*l2^2*f4+8*l2*f0*f5*k1*k3*l1-2*k3*f3*k4*l2*l1-4*l2*f2*f5*k3^2*l1-4*f1*f4 *k3*k2*l1^2-4*f6*k3^2*f0*l1^2-2*f1*k2*k4*l1^2-4*f0*f4*k2^2*l1^2-4*k3*k4*f2*l1^2 -4*f4*k3^2*f2*l1^2: BBB[2,3] := k3*k2*l4^2+l3*k4^2*l2+l4*f3*k3^2*l1-l2*l4*k3*k4-k4*l2*k2*f1*l1+k4*l2^2*k1*f1+k3 *f3*k2*f1*l1^2-k3*f3*k1*f1*l2*l1+l3^2*f3*k1*k4-l2*l4*k1*k2*f1-l3*l4*k3*f3*k1-l3 *l4*k4*k2-k1*k2*f1*l3*f3*l1+l3*k1^2*l2*f3*f1-k3*k4*l3*f3*l1+l4*k2^2*f1*l1+2*l4* f2*k2*k3*l1-2*l4*f1*k1*k3*l1+2*l2*l4*f0*k1^2+2*f0*f3*k2^2*l1^2+4*f0*f4*k2*k3*l1 ^2-4*l2*f0*f4*k1*k3*l1+2*l2*f1*f5*k3^2*l1-4*l2*f0*k2*k1*f3*l1-2*l2*k4*k1*f0*l1+ 2*l2*f0*k3*f5*k2*l1+4*l2*f6*k3^2*f0*l1-2*l2^2*f0*f5*k1*k3+2*f0*f5*k3^2*l1^2+2* f1*l2^2*f6*k3^2+2*f0*k4*k2*l1^2+2*k3*k4*f1*l1^2+2*f4*k3^2*f1*l1^2+2*l2^2*f0*k1^ 2*f3+2*l3^2*f0*f5*k1^2+2*k1^2*f1*l3^2*f4+4*f0*k1^2*l3*l2*f4-2*l2*l4*k3*k1*f2-2* l4*f0*k2*k1*l1+2*l3*l4*k1^2*f1-4*l3*f0*k2*f6*k3*l1-2*l3*f1*f5*k3*k2*l1-4*l3*k3* f4*k1*f1*l1+2*l3*k4*k1*l2*f2-4*l3*l2*f0*f6*k1*k3+2*f0*k2*k1*l2*l3*f5+2*l3^2*f1* f5*k2*k1+4*l3^2*f0*f6*k2*k1-4*l3*f0*f5*k1*k3*l1-2*l3*f0*k2^2*f5*l1-4*l3*f0*f4* k2*k1*l1+2*l3^2*k2^2*f6*f1-4*l3*k3*l2*f6*k2*f1-2*l3*k3*l2*f5*k1*f1-2*l3*k4*k1* f1*l1-2*l3*k2*f2*k4*l1: BBB[2,4] := l2*l4*k4^2+k2*k4*l4^2-k1^2*f3*f1*l3^2*f5-l3*k2*f3^2*k4*l1-k1^2*f3*f1^2*l1^2-k3^ 2*f5^2*f3*l3^2+f1*f3*k2*k4*l1^2-k3^2*f5*f3*f1*l1^2+l3^2*k2*f3*k4*f5+l2*k3*f3*k2 *f5*f1*l1+l3*k3*f3^3*k1*l1+k2*k4*l2^2*f5*f1+l2*l4*k2*f3*k4+l2*l4*k2^2*f5*f1+l2* l4*k1^2*f3*f1+k4*l2*k1*f1*f3*l1+l3*l4*k4*k1*f3+l2*l4*k3^2*f5*f3+l4*k2*f3*k1*f1* l1-l2*l4*k3*f3^2*k1+l3*k3*f3*k4*l2*f5+l4*k3*f3*k4*l1+l3*l4*k3*f5*k2*f3+l3*k2*l2 *f3*k1*f5*f1+2*l4*f0*k2^2*f5*l1+4*l4*f0*k1^2*f3*l1+2*l4*k4*k1*f1*l1+2*l4*k2*f2* k4*l1+4*l3*f0*f4*k1^2*f3*l1+4*l3*k2*f2*k1*f5*f1*l1-8*l3*f0*f2*f6*k2*k1*l1+4*l3^ 2*k3*f3*k1*f6*f2+8*l3^2*f0*f6^2*k3*k2+4*l3^2*k3*f5*k2*f6*f1+2*l3^2*k3*f3^2*k2* f6+4*f5*k3*k1*f1^2*l1^2+4*f0*f4*k3*k1*f3*l1^2+4*k3*f6*k2*f1^2*l1^2+4*f0*k2^2*f6 *f1*l1^2+8*f5*k1^2*f0^2*l1^2+4*f0*k1^2*f3*f2*l1^2+4*f0*k4*k1*f3*l1^2+4*f0*f5*f1 *k2*k1*l1^2-8*f0*f5*k3*k1*f2*l1^2+2*l3*k3^2*f3^2*l2*f6+8*l3*l2*f0*f5*f6*k3*k2+8 *l3*l2*f0*f6^2*k3^2+4*f0*f5^2*k3*k1*l2*l3+4*l3^2*f0*f5^2*k2*k1+4*l3^2*k3*f5^2* k1*f1+4*k3^2*f6*f3*l3^2*f4-8*l3^2*f0*f4*f6*k2*k1+4*l3*k3*l2*f4*k1*f5*f1+4*l3*l2 *f1*k3^2*f6*f5-8*l3^2*k3*f4*k1*f6*f1+8*l3*f0*f4^2*k2*k1*l1+4*l3*f6*k2*k1*f1^2* l1-8*l3*f1*k3^2*f6*f4*l1+4*l3^2*k3*f3*k4*f6+4*l3*k2*f2*k4*f4*l1+4*f0*f4*k2*k1* l2*f3*l1-8*l2*f0*f6*k3^2*f4*l1+4*l3*f0*f4*k2^2*f5*l1-4*l3*f1*f6*k3*k4*l1+4*l3* k3*f5*k4*f2*l1-8*l3*f0*f5*f2*k1^2*l1-8*l3*f0*f5*f3*k2*k1*l1-8*l3*f0*f6*k2^2*f3* l1+4*l3^2*f0*f6*k2^2*f5-12*l2*f0*f6*k2*k3*f3*l1+4*l2*f0*f5*k2*k3*f4*l1+2*l2^2* f6*k2*k1*f1^2+4*l2*f0*f5^2*k3^2*l1+4*f1*f5*k3*k1*l2*f2*l1-8*l3*k3*f3*k1*f5*f1* l1-4*f0*f6*k3*k1*l3*f3*l1-4*l3*f0*f5*k1*k4*l1-4*l3*f0*f6*k2*k4*l1+4*l3*k4*k1*f4 *f1*l1+4*l3*f5*k1^2*f1^2*l1+4*l2^2*f0*f6*k3^2*f5+2*l2^2*f0*f5*f3*k2*k1-8*l2^2* f0*f6*k3*k1*f3+2*l2^2*f0*f5^2*k3*k2+2*k3*f3*k2*l2^2*f6*f1+4*f1*l2^2*f6*k3*k1*f2 +4*l2^2*f0*f5*k3*k1*f4+2*f1*l2^2*f6*k3*k4+8*l2*f1*f0*f6*k2*k1*l1+2*l2^2*f0*f5* k1*k4+4*l2*f1*f5*k1^2*f0*l1+2*l2*f0*k1^2*f3^2*l1+8*l2*k1^2*f6*f0^2*l1+2*l2*k2^2 *f6*f1^2*l1+2*f0*k2^2*f5*l2*f3*l1+2*l2*f5*k2*k1*f1^2*l1+8*l2*f0*f4^2*k3*k1*l1-8 *l2*f0*f5*k3*k1*f3*l1-8*l2*f0*f6*k3*k1*f2*l1+2*f1*l2*f5*k3*k4*l1-4*l2*f0*f6*k3* k4*l1+8*l3^2*f1*k3^2*f6^2+4*l2^2*f0*f6*k2^2*f3+4*l2^2*f1*k1^2*f6*f0+2*f1*l2*f4* k2*k4*l1+4*l2*f0*f4*k1*k4*l1+4*l2*f6*k3*k1*f1^2*l1+4*f1*k3*f6*k2*l2*f2*l1-8*f0* f6*k3*k2*f2*l1^2+2*f0*f3^2*k2*k1*l1^2+8*f0^2*f6*k2*k1*l1^2+2*l3*l4*k3*f5*k4-4* l3*l4*f6*f0*k2*k1+2*l3*l4*k2^2*f6*f1-4*l2*l4*f0*f6*k3*k1+2*l2*l4*k3*f5*k2*f2+4* l2*l4*k3*f4*k1*f2+2*l3*l4*k2*k4*f4+2*l2*l4*k3*k4*f4+2*l2*l4*k4*k1*f2+2*l2*l4*k2 *f4*k1*f1+4*l4*f0*f4*k2*k1*l1+4*l3*k3*f4*k2*f5*f1*l1-4*l3*k3*f3*k1*f4*f2*l1+4* l4*k3*f4*k1*f1*l1-4*l4*f0*f5*k3*k1*l1+4*l3*k2^2*f2*f1*f6*l1+8*l3*f2^2*k3*f6*k2* l1+4*l3*f2*k3^2*f6*f3*l1+4*l3*f1*f5^2*k3^2*l1+4*l3*f0*f5^2*k3*k2*l1+8*l3*k3*f4^ 2*k1*f1*l1-8*l3*f0*f6*k2*k3*f4*l1+8*l3*f2^2*f5*k3*k1*l1+2*l3*k3*l2*f5^2*k2*f1-8 *l3*k3*f3*k2*f1*f6*l1-4*l4*f0*f6*k3*k2*l1+2*l4*k3*f5*k2*f1*l1+4*l3*l4*k3^2*f6* f3-4*l3*l4*k3*f6*k1*f1+4*l3*l4*k3*f5*k1*f2+4*l3*l4*k3*f6*k2*f2+2*l3*l4*k2*f5*k1 *f1-12*l3*l2*f0*f3*f6*k2*k1+4*l3*l2*k1^2*f6*f1^2-8*l3*l2*f2*f0*f6*k1^2+4*l3*l2* f0*f5*f4*k2*k1+4*l3*k3*f3*k2*l2*f6*f2-4*l3*l2*f0*f6*k1*k4+2*l3*k4*l2*k1*f5*f1+4 *l3*k2*f2*k1*l2*f6*f1+2*l3*k2^2*l2*f3*f6*f1+2*l3*l2*f0*f5^2*k2^2-8*l3*k3*f3*k1* l2*f6*f1+4*l3*k3*l2*f6*k4*f2+8*l3*f2^2*k3*f6*k1*l2+2*l3*k2*f2*k4*l2*f5-8*f0*f6* k3*k1*l3*l2*f4: BBB[3,1] := k1*k3*l4^2+l3*k4^2*l1-l3^2*k3^2*f5^2-f1^2*k1^2*l1^2+l4*f1*k1*k2*l1-l2*l4*k4*k2+ l3*k3*f5*k4*l2-k1*k3*l3*f3^2*l1+l3*l4*k3*f5*k2+k1*k3*l2^2*f5*f1+l3*k2^2*f5*f1* l1+l2*f1*k1*k4*l1+l3*l4*k1*k4+2*l3*l4*k3^2*f6+l2*l4*k1*f3*k3+l4*k4*k3*l1+2*l4* k1^2*f0*l1+k1*k3*f3*l3^2*f5-2*l3^2*f1*k1*f6*k2+2*l3^2*k4*k3*f6+4*l3^2*k3^2*f6* f4+l3*k1^2*f3*f1*l1+l3*k3^2*f5*f3*l1+l3*k4*k2*f3*l1+2*l3^2*k3*f6*k2*f3-2*k1^2* f0*l2*l3*f5+4*l2^2*f0*f6*k2^2+2*l2^2*f0*f5*k1*k2+2*k3*l2^2*f6*k2*f1-l3^2*f1*k1^ 2*f5+l3*k2*l2*f5*k1*f1+2*l3*k2^2*l2*f6*f1+2*l3*k3^2*l2*f6*f3+2*f0*k1*k4*l1^2+2* l2*f0*f5*k2^2*l1-2*f0*f5*k3*k2*l1^2+2*f0*f3*k1*k2*l1^2+k1*f3*k3*f1*l1^2-2*l2*f1 *k3^2*f6*l1+2*k1^2*f0*l2*f3*l1+k3*l2*f5*k2*f1*l1+4*f0*k1^2*f2*l1^2-k3^2*f5*f1* l1^2+2*l4*f2*k1*k3*l1+4*k1*k3*l3*f6*f0*l1+2*l3*l4*k1*f4*k3+4*l3*k2*l2*f6*k3*f2+ 2*k1*k3*l3*l2*f5*f2+4*k1*k3*l3*f4*f2*l1+2*l3*k1*f4*k2*f1*l1-4*l3*l2*f0*k1*f6*k2 +2*l3*k1*f2*k4*l1+2*l3*k2*f5*k3*f2*l1+2*l3*k4*k3*f4*l1+4*l2*f0*k1*k2*f4*l1-4*l2 *f0*k3*f6*k2*l1+2*f1*l2*f4*k1*k3*l1: BBB[3,2] := k3*k2*l4^2+l3*k4^2*l2+l4*f3*k3^2*l1-l2*l4*k3*k4-k4*l2*k2*f1*l1+k4*l2^2*k1*f1+k3 *f3*k2*f1*l1^2-k3*f3*k1*f1*l2*l1+l3^2*f3*k1*k4-l2*l4*k1*k2*f1-l3*l4*k3*f3*k1-l3 *l4*k4*k2-k1*k2*f1*l3*f3*l1+l3*k1^2*l2*f3*f1-k3*k4*l3*f3*l1+l4*k2^2*f1*l1+2*l4* f2*k2*k3*l1-2*l4*f1*k1*k3*l1+2*l2*l4*f0*k1^2+2*f0*f3*k2^2*l1^2+4*f0*f4*k2*k3*l1 ^2-4*l2*f0*f4*k1*k3*l1+2*l2*f1*f5*k3^2*l1-4*l2*f0*k2*k1*f3*l1-2*l2*k4*k1*f0*l1+ 2*l2*f0*k3*f5*k2*l1+4*l2*f6*k3^2*f0*l1-2*l2^2*f0*f5*k1*k3+2*f0*f5*k3^2*l1^2+2* f1*l2^2*f6*k3^2+2*f0*k4*k2*l1^2+2*k3*k4*f1*l1^2+2*f4*k3^2*f1*l1^2+2*l2^2*f0*k1^ 2*f3+2*l3^2*f0*f5*k1^2+2*k1^2*f1*l3^2*f4+4*f0*k1^2*l3*l2*f4-2*l2*l4*k3*k1*f2-2* l4*f0*k2*k1*l1+2*l3*l4*k1^2*f1-4*l3*f0*k2*f6*k3*l1-2*l3*f1*f5*k3*k2*l1-4*l3*k3* f4*k1*f1*l1+2*l3*k4*k1*l2*f2-4*l3*l2*f0*f6*k1*k3+2*f0*k2*k1*l2*l3*f5+2*l3^2*f1* f5*k2*k1+4*l3^2*f0*f6*k2*k1-4*l3*f0*f5*k1*k3*l1-2*l3*f0*k2^2*f5*l1-4*l3*f0*f4* k2*k1*l1+2*l3^2*k2^2*f6*f1-4*l3*k3*l2*f6*k2*f1-2*l3*k3*l2*f5*k1*f1-2*l3*k4*k1* f1*l1-2*l3*k2*f2*k4*l1: BBB[3,3] := k3^2*l4^2+l3^2*k4^2+k1^2*f1^2*l2^2+k2^2*f1^2*l1^2-4*l3*l4*k1^2*f0-2*l3*l4*k4*k3 +2*k1*l2*l4*k3*f1-4*l4*f0*k2^2*l1-4*f0*k2^2*l3^2*f6-4*k1^2*f0*l3^2*f4-4*k1*f0* k2*l3^2*f5+4*l3*k4*k1*f0*l1+8*l3*f0*f4*k1*k3*l1+4*l3*f0*k3*f5*k2*l1+2*l3*f1*k2* k4*l1+4*k1*l2*l4*f0*k2+4*l4*f0*k1*k3*l1-2*l4*k2*f1*k3*l1+4*l3*l2*f0*f5*k1*k3-4* l3*l2*f0*f3*k1^2-2*l3*k1*f1*k4*l2+8*l3*l2*f0*f6*k2*k3+4*k1*f0*k2*l3*f3*l1+8*l2* f0*k1*f2*k2*l1-2*k1*f1^2*l2*k2*l1+4*f0*k1*k3*l2*f3*l1+4*l2*f0*k4*k2*l1-4*l2*f0* f5*k3^2*l1-4*l2^2*f6*k3^2*f0-4*l2^2*k4*k1*f0-4*f0*f3*k2*k3*l1^2-4*l2^2*k1^2*f2* f0-4*f0*f4*k3^2*l1^2-4*f0*k4*k3*l1^2-4*f0*k2^2*f2*l1^2: BBB[3,4] := l3*l4*k4^2+k3*k4*l4^2-l4*f1*k2*k4*l1+l3*l4*f5*k3^2*f3-l2*l4*k1*f1*k4-l2*l4*k3* f5*k2*f1-l4*k1*f3*k3*f1*l1+k3*k4*f3*l3^2*f5-l3*k3*l2*f3*k1*f5*f1-k3*f5*k2*f1*l3 *f3*l1-k1*f1*k4*l3*f3*l1-l3*k2*f1*k4*l2*f5+k1*f1^2*l2^2*k2*f5+k2^2*f1^2*l2*f5* l1+k1^2*f1^2*l2*f3*l1+k2*f1^2*k1*f3*l1^2-6*l4*k4*k1*f0*l1-8*l4*k1^2*f2*f0*l1-2* l2*l4*f1*k3^2*f6-2*l2*l4*f0*k1^2*f3-2*l2*l4*f0*f5*k2^2-4*l2*l4*f0*k2*k1*f4-2*l4 *f0*k3*f5*k2*l1-4*l4*f4*f0*k3*k1*l1-4*l4*f6*k3^2*f0*l1-4*l3*l4*f0*f5*k2*k1-8*l3 *l4*f0*f6*k2^2-4*l2*l4*f0*f6*k3*k2-2*l2*l4*k3*f4*k1*f1-4*l4*f0*k2*k1*f3*l1-4*l3 ^2*f0*f5^2*k3*k1-4*l3^2*k3*f3*k1*f6*f1-8*l3^2*k3*f4*k2*f6*f1+8*l3^2*f0*f4*f6*k3 *k1-4*l3^2*f0*f5*f6*k3*k2+4*l3*l4*f0*f6*k3*k1-6*l3*l4*k2*f1*k3*f6+2*l3*l4*k3*f4 *k4-2*l3*l4*k1*f1*k3*f5-4*l3*l4*f2*f6*k3^2+2*l4*f1^2*k1^2*l1-4*k3*k4*l3^2*f6*f2 -4*l3^2*f0*f6*k1*k4-4*l3*f0*f5^2*k3^2*l1-2*l3*f6*k2^2*f1^2*l1-2*l3^2*k2*f1*k4* f6-4*l3*l2*f0*f5*f4*k3*k1-4*l3*f0*f4*k1*k4*l1-2*k3*k4*l3*f5*f1*l1-2*l3*k2*f1*k4 *f4*l1+8*l3*f0*f5*k3*k1*f3*l1+4*k3*k4*l3*f6*f0*l1-4*l3*f1*k3^2*f6*f3*l1+2*l3^2* k3^2*f3^2*f6+4*l3*f0*f6*f3*k3*k2*l1+8*l3*f0*f4*f6*k3^2*l1-8*l3*f0*f4^2*k3*k1*l1 -8*l3*f0*f2*f6*k3*k1*l1-4*f0*k2*k1*f4*l3*f3*l1-2*f0*k1^2*f3^2*l3*l1-2*l3*k2*f1^ 2*k1*f5*l1-4*l3*f1*f5*f0*k1^2*l1-2*f0*f5*k2^2*l3*f3*l1+4*l3*f6*k3*k1*f1^2*l1-4* l3*l2*f0*f5*f3*k2*k1-4*l3*f0*f4*f5*k3*k2*l1+2*l3^2*f2*f5^2*k3^2-4*l3*f6*k1*f1* l2*k3*f2-8*l3*l2*f0*f6*k2^2*f3-2*l3*f6*k1*f1^2*l2*k2-6*k3*k4*l3*l2*f6*f1-4*l3* l2*f0*f6*k2*k4+8*l3*k1^2*f6*f0^2*l1+2*l3*l2*f1*f5^2*k3^2-4*l3*l2*f0*f5*f6*k3^2- 16*l3*l2*f0*f4*f6*k3*k2-8*k1^2*f2*f0*l3^2*f6+2*f0*k1^2*f3*l3^2*f5-8*l3^2*f0*f6^ 2*k3^2-8*l3^2*f2*f6*k3^2*f4-4*l3*f6*k2*f1*k3*l2*f3+2*l3^2*k3*f5^2*k2*f1-8*f0*f6 *k2^2*l3^2*f4+2*f0*f5^2*k2^2*l3^2-8*l2^2*f0*f6*f3*k3*k2-4*f0*k2*k1*f3*l3^2*f6+2 *f0*k3*f5^2*k2*l2*l3+4*l3*l2*f0*f3*f6*k3*k1-2*l3*l2*f0*f5*k1*k4-8*l3*l2*f1*k3^2 *f6*f4-4*l3*l2*f1*f0*f6*k1^2+2*f1^2*k1^2*l3^2*f6-2*l2^2*f0*f5*k3*k1*f3-2*l2^2* f6*k3*k1*f1^2-4*l2*f0*f6*k2^2*f1*l1-4*l2*f0*f5*k2^2*f2*l1-4*l2*f1*f5*f0*k2*k1* l1+2*k2*f1^2*k1*l2*f4*l1-4*l2*f0*f3*k1^2*f2*l1-8*l2^2*f0*f6*k3*k4-8*l2*f0^2*f6* k2*k1*l1-2*l2^2*f0*f5*k2*k4+2*l2^2*f0*f5^2*k3^2-8*l2^2*f0*f4*f6*k3^2-2*l2*f5*k3 *k1*f1^2*l1-4*l2*f0*f4*k2*k4*l1-4*l2*f0*f5*k3*k4*l1-4*l2*f0*f3*k1*k4*l1-4*f6*k3 ^2*f0*l2*f3*l1-8*l2*f0*f4*f2*k2*k1*l1-8*l2*f5*k1^2*f0^2*l1-4*l2^2*f2*f5*f0*k2* k1-8*l2^2*k1^2*f6*f0^2-2*l2*k3*f6*k2*f1^2*l1-4*l2*f0*f5*f3*k3*k2*l1-4*f4*f0*k3* k1*l2*f3*l1-8*l2^2*f0*f6*k2^2*f2-4*l2^2*f1*f0*f6*k2*k1+8*f0^2*f6*k3*k1*l1^2-4* f0*k3*f6*k2*f1*l1^2+2*k2^2*f1^2*l2^2*f6-8*f2*f6*k3^2*f0*l1^2-4*f0*f5*f1*k3*k1* l1^2-2*f0*f3^2*k3*k1*l1^2+2*f0*f3*f5*k3^2*l1^2-2*f0*f3*k2*k4*l1^2+2*k1*f1^2*k4* l1^2+2*f1^2*k3^2*f6*l1^2-8*f5*f0^2*k2*k1*l1^2-4*f0*k2*k1*f3*f2*l1^2-8*k1^2*f2^2 *f0*l1^2+2*f1^2*k1^2*f2*l1^2+4*f0*f3*k1^2*f1*l1^2-8*f4*k1^2*f0^2*l1^2-8*f0*f2* k1*k4*l1^2-8*f0^2*f6*k2^2*l1^2-4*l3*f6*k2*f1*k3*f2*l1-4*l3*k3*f2*k1*f5*f1*l1: BBB[4,1] := k1*k4*l4^2+l4*k4^2*l1-l2*k4*k2*f5*f1*l1-k3*l2*f5*k1*f1*f3*l1-l3*l4*k1*f3*k3*f5+ k1*k4*f1*f3*l1^2-l3*l4*k2*f5*k4-l2*l4*f1*k1*k2*f5+l3^2*k2*f5^2*k3*f3+l3*k2^2*l2 *f5^2*f1-l2*l4*k4*k3*f5+l4*k1^2*f3*f1*l1-f1*k1*k2*f5*l3*f3*l1+l3*k3^2*f5^2*l2* f3-k4*k3*f5*l3*f3*l1+k3*l2^2*f5^2*k2*f1+2*l3*l4*k3^2*f5^2+2*f0*f5^2*k1^2*l3^2-4 *l2*f1*f6*k1^2*f0*l1+2*l2*f5*k1^2*f1^2*l1+2*l3^2*k3*f5^2*k4-4*l2*f0*f5*f6*k3^2* l1-8*l2^2*f0*k1*f6*k4+4*f0*f6*k1*k3*l2*f3*l1-4*l2*f0*f6*k2*k4*l1-4*f1*k3*f6*k1* l2*f2*l1-6*l2*f0*f5*k1*k4*l1-2*k4*k2*l2^2*f6*f1-2*k3*l2^2*f6*k1*f1*f3-2*l2^2*f0 *f5^2*k1*k3-4*k3*f6*k2*f1*l2*f3*l1-4*l2*f0*f5*f4*k1*k3*l1-8*f0*f6*k2^2*l2*f3*l1 -4*l2*f0*f5*f3*k1*k2*l1-2*l2*f0*f5^2*k3*k2*l1+2*l2*f6*k1*k2*f1^2*l1-8*l2*f0*f5* k1^2*f2*l1-16*l2*f0*f6*k1*k2*f2*l1+2*l2^2*f6*k1^2*f1^2-2*f1*l2*f6*k3*k4*l1-8*f0 *f5*k1*k2*f2*l1^2+2*f1^2*k1*k2*f5*l1^2-4*f0*k1*f6*k2*f1*l1^2+2*k1^2*f4*f1^2*l1^ 2-8*f0*f6*k2^2*f2*l1^2+8*f0*f2*f6*k1*k3*l1^2-4*k3*f6*k2*f3*f0*l1^2+2*k3^2*f6*f3 *f1*l1^2+2*k2^2*f6*f1^2*l1^2-8*k3^2*f6*f4*f0*l1^2-4*f0*k3*f6*k4*l1^2-4*f1^2*f6* k1*k3*l1^2-4*f0*f3*f5*k1*k3*l1^2-8*f0*f4*k1^2*f2*l1^2+2*f0*f3^2*k1^2*l1^2-8*f6* k1^2*f0^2*l1^2-4*f0*f4*k1*k4*l1^2-2*f0*f5*k2*k4*l1^2+2*k3^2*f5^2*f0*l1^2-4*l2* l4*f0*k1*f6*k2-4*l2*l4*k3*f6*k2*f2-4*l3*l4*k1*f2*k3*f6-2*l3*l4*f1*k1*f6*k2-6*l3 *l4*k4*k3*f6+2*l4*k1*f2*k4*l1-8*l3*l4*k3^2*f6*f4-4*l4*f4*k1^2*f0*l1-4*l4*k3*f6* k2*f1*l1-8*l4*f0*f6*k2^2*l1-2*l4*k3*k1*f5*f1*l1-4*l3*k3*l2*f5*k2*f6*f1-8*k3*f6* k2*f2*l3*l2*f4-2*l2*l4*k1*f2*k3*f5-2*l2*l4*k3^2*f6*f3-6*l4*f0*k1*k2*f5*l1-4*l3* k1*l2*f3*k2*f6*f1-2*l2*l4*f0*f5*k1^2+4*l4*f0*f6*k1*k3*l1-4*l3*l4*k3*f6*k2*f3-2* l2*l4*k2^2*f6*f1-4*l3*l4*f6*k1^2*f0-4*l3^2*f0*f5*k1*f6*k2-8*k3^2*f6*f4^2*l3^2-4 *k3*f6*k2*f3*l3^2*f4+4*l3^2*k3^2*f5*f6*f3-8*l3^2*k3*f6^2*k2*f1+2*k3^2*f5^2*l3^2 *f4-8*f0*f6^2*k2^2*l3^2+2*l3^2*k1^2*f3*f6*f1-8*f6*k1^2*f0*l3^2*f4-4*k2^2*f6*f1* l3*l2*f4-4*l3*l2*f0*f5*f6*k2^2-4*l3*l2*f0*k1^2*f6*f3-2*l3*l2*f0*f5^2*k1*k2+8*l3 *k1*f3*k3*f1*f6*l1-2*l3^2*k4*k2*f6*f3-2*k1*k4*l3*f5*f1*l1-2*l3*k4*k2*f5*f2*l1-8 *l3^2*k3*f6*k4*f4+8*l3^2*f0*k1*f6^2*k3-4*l3^2*k3*f5*k1*f6*f1-2*l3^2*k1*f3^2*k3* f6-8*l3^2*f2*k3^2*f6^2+4*l3*f0*f6*f3*k1*k2*l1-8*l3*f0*f4*f6*k1*k3*l1+4*l3*f0*f5 ^2*k1*k3*l1-2*l3*f0*f5^2*k2^2*l1-8*l3*f2^2*k3*f6*k1*l1-4*l3*f1*k3^2*f6*f5*l1-4* l3*k3*f5*k1*f4*f1*l1+4*k1*k4*l3*f6*f0*l1+8*l3*f0*f6^2*k3^2*l1-2*k3^2*f6*f3^2*l3 *l1-4*l3*f0*f4*k1*k2*f5*l1-2*k2^2*f6*f1*l3*f3*l1-4*k3*f6*k2*f2*l3*f3*l1-2*l3*k3 *f5^2*k2*f1*l1-4*l3*k1*f2*k2*f1*f6*l1+8*l3*f2*f6*k1^2*f0*l1-4*l3*f0*f5*k1^2*f3* l1-4*l3*k1*f2*k3*l2*f6*f3-2*l3*k3*l2*f5^2*k1*f1-4*l3*k4*k3*f2*f6*l1-4*k3^2*f6* f3*l3*l2*f4-8*l3*l2*f1*f6^2*k3^2+2*l3*k3*f5^2*k2*l2*f2-4*k1*k4*l3*l2*f6*f1-4*l3 *k4*k2*l2*f6*f2-4*l3*k4*k3*l2*f6*f3-4*l3*f1^2*k1^2*f6*l1-8*f0*f6*k2^2*l2^2*f4-8 *l2^2*f2*f6*k1^2*f0-8*l3*l2*f0*f6^2*k3*k2-8*l2^2*f0*f6*f3*k1*k2+2*l2^2*f0*f5^2* k2^2-4*k3*f6*k2*f1*l2^2*f4-8*l2^2*f0*f6^2*k3^2-4*l2^2*f0*f6*k3*k2*f5: BBB[4,2] := l2*l4*k4^2+k2*k4*l4^2-k1^2*f3*f1*l3^2*f5-l3*k2*f3^2*k4*l1-k1^2*f3*f1^2*l1^2-k3^ 2*f5^2*f3*l3^2+f1*f3*k2*k4*l1^2-k3^2*f5*f3*f1*l1^2+l3^2*k2*f3*k4*f5+l2*k3*f3*k2 *f5*f1*l1+l3*k3*f3^3*k1*l1+k2*k4*l2^2*f5*f1+l2*l4*k2*f3*k4+l2*l4*k2^2*f5*f1+l2* l4*k1^2*f3*f1+k4*l2*k1*f1*f3*l1+l3*l4*k4*k1*f3+l2*l4*k3^2*f5*f3+l4*k2*f3*k1*f1* l1-l2*l4*k3*f3^2*k1+l3*k3*f3*k4*l2*f5+l4*k3*f3*k4*l1+l3*l4*k3*f5*k2*f3+l3*k2*l2 *f3*k1*f5*f1+2*l4*f0*k2^2*f5*l1+4*l4*f0*k1^2*f3*l1+2*l4*k4*k1*f1*l1+2*l4*k2*f2* k4*l1+4*l3*f0*f4*k1^2*f3*l1+4*l3*k2*f2*k1*f5*f1*l1-8*l3*f0*f2*f6*k2*k1*l1+4*l3^ 2*k3*f3*k1*f6*f2+8*l3^2*f0*f6^2*k3*k2+4*l3^2*k3*f5*k2*f6*f1+2*l3^2*k3*f3^2*k2* f6+4*f5*k3*k1*f1^2*l1^2+4*f0*f4*k3*k1*f3*l1^2+4*k3*f6*k2*f1^2*l1^2+4*f0*k2^2*f6 *f1*l1^2+8*f5*k1^2*f0^2*l1^2+4*f0*k1^2*f3*f2*l1^2+4*f0*k4*k1*f3*l1^2+4*f0*f5*f1 *k2*k1*l1^2-8*f0*f5*k3*k1*f2*l1^2+2*l3*k3^2*f3^2*l2*f6+8*l3*l2*f0*f5*f6*k3*k2+8 *l3*l2*f0*f6^2*k3^2+4*f0*f5^2*k3*k1*l2*l3+4*l3^2*f0*f5^2*k2*k1+4*l3^2*k3*f5^2* k1*f1+4*k3^2*f6*f3*l3^2*f4-8*l3^2*f0*f4*f6*k2*k1+4*l3*k3*l2*f4*k1*f5*f1+4*l3*l2 *f1*k3^2*f6*f5-8*l3^2*k3*f4*k1*f6*f1+8*l3*f0*f4^2*k2*k1*l1+4*l3*f6*k2*k1*f1^2* l1-8*l3*f1*k3^2*f6*f4*l1+4*l3^2*k3*f3*k4*f6+4*l3*k2*f2*k4*f4*l1+4*f0*f4*k2*k1* l2*f3*l1-8*l2*f0*f6*k3^2*f4*l1+4*l3*f0*f4*k2^2*f5*l1-4*l3*f1*f6*k3*k4*l1+4*l3* k3*f5*k4*f2*l1-8*l3*f0*f5*f2*k1^2*l1-8*l3*f0*f5*f3*k2*k1*l1-8*l3*f0*f6*k2^2*f3* l1+4*l3^2*f0*f6*k2^2*f5-12*l2*f0*f6*k2*k3*f3*l1+4*l2*f0*f5*k2*k3*f4*l1+2*l2^2* f6*k2*k1*f1^2+4*l2*f0*f5^2*k3^2*l1+4*f1*f5*k3*k1*l2*f2*l1-8*l3*k3*f3*k1*f5*f1* l1-4*f0*f6*k3*k1*l3*f3*l1-4*l3*f0*f5*k1*k4*l1-4*l3*f0*f6*k2*k4*l1+4*l3*k4*k1*f4 *f1*l1+4*l3*f5*k1^2*f1^2*l1+4*l2^2*f0*f6*k3^2*f5+2*l2^2*f0*f5*f3*k2*k1-8*l2^2* f0*f6*k3*k1*f3+2*l2^2*f0*f5^2*k3*k2+2*k3*f3*k2*l2^2*f6*f1+4*f1*l2^2*f6*k3*k1*f2 +4*l2^2*f0*f5*k3*k1*f4+2*f1*l2^2*f6*k3*k4+8*l2*f1*f0*f6*k2*k1*l1+2*l2^2*f0*f5* k1*k4+4*l2*f1*f5*k1^2*f0*l1+2*l2*f0*k1^2*f3^2*l1+8*l2*k1^2*f6*f0^2*l1+2*l2*k2^2 *f6*f1^2*l1+2*f0*k2^2*f5*l2*f3*l1+2*l2*f5*k2*k1*f1^2*l1+8*l2*f0*f4^2*k3*k1*l1-8 *l2*f0*f5*k3*k1*f3*l1-8*l2*f0*f6*k3*k1*f2*l1+2*f1*l2*f5*k3*k4*l1-4*l2*f0*f6*k3* k4*l1+8*l3^2*f1*k3^2*f6^2+4*l2^2*f0*f6*k2^2*f3+4*l2^2*f1*k1^2*f6*f0+2*f1*l2*f4* k2*k4*l1+4*l2*f0*f4*k1*k4*l1+4*l2*f6*k3*k1*f1^2*l1+4*f1*k3*f6*k2*l2*f2*l1-8*f0* f6*k3*k2*f2*l1^2+2*f0*f3^2*k2*k1*l1^2+8*f0^2*f6*k2*k1*l1^2+2*l3*l4*k3*f5*k4-4* l3*l4*f6*f0*k2*k1+2*l3*l4*k2^2*f6*f1-4*l2*l4*f0*f6*k3*k1+2*l2*l4*k3*f5*k2*f2+4* l2*l4*k3*f4*k1*f2+2*l3*l4*k2*k4*f4+2*l2*l4*k3*k4*f4+2*l2*l4*k4*k1*f2+2*l2*l4*k2 *f4*k1*f1+4*l4*f0*f4*k2*k1*l1+4*l3*k3*f4*k2*f5*f1*l1-4*l3*k3*f3*k1*f4*f2*l1+4* l4*k3*f4*k1*f1*l1-4*l4*f0*f5*k3*k1*l1+4*l3*k2^2*f2*f1*f6*l1+8*l3*f2^2*k3*f6*k2* l1+4*l3*f2*k3^2*f6*f3*l1+4*l3*f1*f5^2*k3^2*l1+4*l3*f0*f5^2*k3*k2*l1+8*l3*k3*f4^ 2*k1*f1*l1-8*l3*f0*f6*k2*k3*f4*l1+8*l3*f2^2*f5*k3*k1*l1+2*l3*k3*l2*f5^2*k2*f1-8 *l3*k3*f3*k2*f1*f6*l1-4*l4*f0*f6*k3*k2*l1+2*l4*k3*f5*k2*f1*l1+4*l3*l4*k3^2*f6* f3-4*l3*l4*k3*f6*k1*f1+4*l3*l4*k3*f5*k1*f2+4*l3*l4*k3*f6*k2*f2+2*l3*l4*k2*f5*k1 *f1-12*l3*l2*f0*f3*f6*k2*k1+4*l3*l2*k1^2*f6*f1^2-8*l3*l2*f2*f0*f6*k1^2+4*l3*l2* f0*f5*f4*k2*k1+4*l3*k3*f3*k2*l2*f6*f2-4*l3*l2*f0*f6*k1*k4+2*l3*k4*l2*k1*f5*f1+4 *l3*k2*f2*k1*l2*f6*f1+2*l3*k2^2*l2*f3*f6*f1+2*l3*l2*f0*f5^2*k2^2-8*l3*k3*f3*k1* l2*f6*f1+4*l3*k3*l2*f6*k4*f2+8*l3*f2^2*k3*f6*k1*l2+2*l3*k2*f2*k4*l2*f5-8*f0*f6* k3*k1*l3*l2*f4: BBB[4,3] := l3*l4*k4^2+k3*k4*l4^2-l4*f1*k2*k4*l1+l3*l4*f5*k3^2*f3-l2*l4*k1*f1*k4-l2*l4*k3* f5*k2*f1-l4*k1*f3*k3*f1*l1+k3*k4*f3*l3^2*f5-l3*k3*l2*f3*k1*f5*f1-k3*f5*k2*f1*l3 *f3*l1-k1*f1*k4*l3*f3*l1-l3*k2*f1*k4*l2*f5+k1*f1^2*l2^2*k2*f5+k2^2*f1^2*l2*f5* l1+k1^2*f1^2*l2*f3*l1+k2*f1^2*k1*f3*l1^2-6*l4*k4*k1*f0*l1-8*l4*k1^2*f2*f0*l1-2* l2*l4*f1*k3^2*f6-2*l2*l4*f0*k1^2*f3-2*l2*l4*f0*f5*k2^2-4*l2*l4*f0*k2*k1*f4-2*l4 *f0*k3*f5*k2*l1-4*l4*f4*f0*k3*k1*l1-4*l4*f6*k3^2*f0*l1-4*l3*l4*f0*f5*k2*k1-8*l3 *l4*f0*f6*k2^2-4*l2*l4*f0*f6*k3*k2-2*l2*l4*k3*f4*k1*f1-4*l4*f0*k2*k1*f3*l1-4*l3 ^2*f0*f5^2*k3*k1-4*l3^2*k3*f3*k1*f6*f1-8*l3^2*k3*f4*k2*f6*f1+8*l3^2*f0*f4*f6*k3 *k1-4*l3^2*f0*f5*f6*k3*k2+4*l3*l4*f0*f6*k3*k1-6*l3*l4*k2*f1*k3*f6+2*l3*l4*k3*f4 *k4-2*l3*l4*k1*f1*k3*f5-4*l3*l4*f2*f6*k3^2+2*l4*f1^2*k1^2*l1-4*k3*k4*l3^2*f6*f2 -4*l3^2*f0*f6*k1*k4-4*l3*f0*f5^2*k3^2*l1-2*l3*f6*k2^2*f1^2*l1-2*l3^2*k2*f1*k4* f6-4*l3*l2*f0*f5*f4*k3*k1-4*l3*f0*f4*k1*k4*l1-2*k3*k4*l3*f5*f1*l1-2*l3*k2*f1*k4 *f4*l1+8*l3*f0*f5*k3*k1*f3*l1+4*k3*k4*l3*f6*f0*l1-4*l3*f1*k3^2*f6*f3*l1+2*l3^2* k3^2*f3^2*f6+4*l3*f0*f6*f3*k3*k2*l1+8*l3*f0*f4*f6*k3^2*l1-8*l3*f0*f4^2*k3*k1*l1 -8*l3*f0*f2*f6*k3*k1*l1-4*f0*k2*k1*f4*l3*f3*l1-2*f0*k1^2*f3^2*l3*l1-2*l3*k2*f1^ 2*k1*f5*l1-4*l3*f1*f5*f0*k1^2*l1-2*f0*f5*k2^2*l3*f3*l1+4*l3*f6*k3*k1*f1^2*l1-4* l3*l2*f0*f5*f3*k2*k1-4*l3*f0*f4*f5*k3*k2*l1+2*l3^2*f2*f5^2*k3^2-4*l3*f6*k1*f1* l2*k3*f2-8*l3*l2*f0*f6*k2^2*f3-2*l3*f6*k1*f1^2*l2*k2-6*k3*k4*l3*l2*f6*f1-4*l3* l2*f0*f6*k2*k4+8*l3*k1^2*f6*f0^2*l1+2*l3*l2*f1*f5^2*k3^2-4*l3*l2*f0*f5*f6*k3^2- 16*l3*l2*f0*f4*f6*k3*k2-8*k1^2*f2*f0*l3^2*f6+2*f0*k1^2*f3*l3^2*f5-8*l3^2*f0*f6^ 2*k3^2-8*l3^2*f2*f6*k3^2*f4-4*l3*f6*k2*f1*k3*l2*f3+2*l3^2*k3*f5^2*k2*f1-8*f0*f6 *k2^2*l3^2*f4+2*f0*f5^2*k2^2*l3^2-8*l2^2*f0*f6*f3*k3*k2-4*f0*k2*k1*f3*l3^2*f6+2 *f0*k3*f5^2*k2*l2*l3+4*l3*l2*f0*f3*f6*k3*k1-2*l3*l2*f0*f5*k1*k4-8*l3*l2*f1*k3^2 *f6*f4-4*l3*l2*f1*f0*f6*k1^2+2*f1^2*k1^2*l3^2*f6-2*l2^2*f0*f5*k3*k1*f3-2*l2^2* f6*k3*k1*f1^2-4*l2*f0*f6*k2^2*f1*l1-4*l2*f0*f5*k2^2*f2*l1-4*l2*f1*f5*f0*k2*k1* l1+2*k2*f1^2*k1*l2*f4*l1-4*l2*f0*f3*k1^2*f2*l1-8*l2^2*f0*f6*k3*k4-8*l2*f0^2*f6* k2*k1*l1-2*l2^2*f0*f5*k2*k4+2*l2^2*f0*f5^2*k3^2-8*l2^2*f0*f4*f6*k3^2-2*l2*f5*k3 *k1*f1^2*l1-4*l2*f0*f4*k2*k4*l1-4*l2*f0*f5*k3*k4*l1-4*l2*f0*f3*k1*k4*l1-4*f6*k3 ^2*f0*l2*f3*l1-8*l2*f0*f4*f2*k2*k1*l1-8*l2*f5*k1^2*f0^2*l1-4*l2^2*f2*f5*f0*k2* k1-8*l2^2*k1^2*f6*f0^2-2*l2*k3*f6*k2*f1^2*l1-4*l2*f0*f5*f3*k3*k2*l1-4*f4*f0*k3* k1*l2*f3*l1-8*l2^2*f0*f6*k2^2*f2-4*l2^2*f1*f0*f6*k2*k1+8*f0^2*f6*k3*k1*l1^2-4* f0*k3*f6*k2*f1*l1^2+2*k2^2*f1^2*l2^2*f6-8*f2*f6*k3^2*f0*l1^2-4*f0*f5*f1*k3*k1* l1^2-2*f0*f3^2*k3*k1*l1^2+2*f0*f3*f5*k3^2*l1^2-2*f0*f3*k2*k4*l1^2+2*k1*f1^2*k4* l1^2+2*f1^2*k3^2*f6*l1^2-8*f5*f0^2*k2*k1*l1^2-4*f0*k2*k1*f3*f2*l1^2-8*k1^2*f2^2 *f0*l1^2+2*f1^2*k1^2*f2*l1^2+4*f0*f3*k1^2*f1*l1^2-8*f4*k1^2*f0^2*l1^2-8*f0*f2* k1*k4*l1^2-8*f0^2*f6*k2^2*l1^2-4*l3*f6*k2*f1*k3*f2*l1-4*l3*k3*f2*k1*f5*f1*l1: BBB[4,4] := k4^2*l4^2+k2^2*l2^2*f5^2*f1^2+k1^2*f1^2*f3^2*l1^2+l3^2*k3^2*f3^2*f5^2-2*f5*k1^2 *f1^3*l1^2+16*l2*f0*f6*k2*f2*k4*l1+16*f4*f0*f6*k3^2*l2*f3*l1-4*k2^2*f6*f1^2*l2* f3*l1+16*l2*f0*f3*f6*k3*k4*l1-4*l2*f5*k1*f1^2*k4*l1-16*l2*f1*f0*f6*k3*k1*f4*l1+ 4*l2*f0*f5*k3*f3^2*k1*l1+16*l2^2*f2*f0*f6^2*k3^2-4*k2^2*f6*f1^2*l2^2*f4-4*k1*f1 ^2*f3*k2*l2^2*f6+16*l2^2*f0^2*f6*k2*f5*k1-4*l2^2*f2*f5^2*f0*k2^2-4*l2*f6*k2*f1^ 2*k4*l1+16*l2*f1*f0*f6^2*k3^2*l1+8*l2^2*f1*f0*f6*k3*f5*k1+16*l2^2*f1*f0*f6^2*k3 *k2-4*l2^2*f0*f5^2*f3*k3*k2-8*l2^2*f0*f5*f3*f6*k3^2+16*l3^2*f2^2*k3^2*f6^2-4*f0 *f5^2*k3^2*l2^2*f4+16*f4^2*f0*f6*k3^2*l2^2+8*l2^2*f0*f6*k3*f3^2*k1-8*l2^2*f1*f0 *f6*k1^2*f3+16*l2^2*f0^2*f6*k1^2*f4-4*l2^2*f2*f6*k1^2*f1^2-2*f5*k2*f1^2*k4*l1^2 +16*l2^2*f0*f6*k1^2*f2^2+16*f3*f0*f6*k3*k2*l2^2*f4+8*f0*f5*k2*f2*k4*l1^2+16*l2^ 2*f0^2*f6^2*k2^2+16*f0^2*f6*f4*k2^2*l1^2-4*k2^2*f6*f1^2*f2*l1^2+16*f0*f6*k2*f1* k1*f2*l1^2+16*l3^2*f0^2*f6^2*k1^2+16*f3*f0*f6*k3*k2*f2*l1^2-8*f0*f6*k2^2*f3*f1* l1^2-4*k3^2*f6*f3^2*f0*l1^2-8*f1^2*k3^2*f6*f4*l1^2+16*f0^2*f6^2*k3^2*l1^2-4*f1^ 2*f6*k3*k4*l1^2-2*f5*k3*f3*k1*f1^2*l1^2-32*f0^2*f6*k3*k1*f4*l1^2+16*f5*f0^2*f6* k3*k2*l1^2+16*f5^2*f0^2*k3*k1*l1^2+8*f3*f0*k3*f5*k1*f2*l1^2-4*k2*f6*f1^2*k3*f3* l1^2-8*f2*f5^2*k3^2*f0*l1^2+16*f2*f6*k3*k4*f0*l1^2+16*f0*f4*f2*k1*k4*l1^2-4*f0* f3^2*k1*k4*l1^2+32*f2*k3^2*f6*f4*f0*l1^2+8*f0*f6*k3*f3*k1*f1*l1^2+8*f0*f5*k1^2* f1*f2*l1^2-16*f5*f0^2*k1^2*f3*l1^2-4*f3^2*f0*k1^2*f2*l1^2-8*f0*f4*f1*k1^2*f3*l1 ^2-4*f4*k1^2*f1^2*f2*l1^2+16*f2^2*f0*f6*k2^2*l1^2+16*f4^2*f0^2*k1^2*l1^2+16*f0^ 2*f6*k1^2*f2*l1^2-4*k1^2*f6*f1^2*f0*l1^2+16*f2^2*f0*k1^2*f4*l1^2-4*f4*k1*f1^2* k4*l1^2+2*f1^2*k3^2*f5^2*l1^2+16*f2^2*f0*k2*f5*k1*l1^2-4*k2*f6*f1^3*k1*l1^2-4* f5*k2*f1^2*k1*f2*l1^2-16*f0^2*f6*k2*k1*f3*l1^2+16*f4*f0^2*k2*f5*k1*l1^2+4*l2*l4 *f0*f5*k1*f3*k2+16*l3*l4*f0*f6*k1^2*f2-4*l3*l4*k1^2*f6*f1^2+16*l2*l4*f0*k1*f6* f2*k2+8*l3*l4*f2*f6*k3*k4-2*l3*l4*f5*k3*f3*k4+16*l3*l4*f2*k3^2*f6*f4+8*l3*l4*f0 *f6*k1*k4-4*l3*l4*k3^2*f6*f3^2-4*l3*l4*f5^2*k2*f1*k3-4*l3*l4*f2*f5^2*k3^2+16*l3 *l4*f0*f6*f4*k2^2+16*l3*l4*f0*f6*k2*k1*f3+8*l2*l4*f0*k4*f6*k2+8*l2*l4*f0*k1*f6* f3*k3+16*l4*f2*f0*k2*f5*k1*l1+4*l3*l4*k4*k2*f1*f6-4*l3*l4*f0*f5^2*k2^2+8*l3*l4* f6*k3*f3*k1*f1+16*l3*l4*k2*f6*f1*k3*f4+4*l4*f0*k2*f5*k4*l1+16*l4*f3*f0*f6*k3*k2 *l1+16*l4*f2*f0*f6*k2^2*l1-4*l4*f4*k1^2*f1^2*l1+16*l4*f2*f0*k1^2*f4*l1-4*l2*l4* f0*k2*f5^2*k3+4*l2*l4*f1*f6*k3*k4-2*l2*l4*k1^2*f5*f1^2-4*l4*k2^2*f6*f1^2*l1+16* l4*f4*f0*f6*k3^2*l1+8*l4*f0*f6*k3*k4*l1+8*l4*f3*f0*k3*f5*k1*l1-4*l4*f0*f5^2*k3^ 2*l1-4*l4*f3^2*f0*k1^2*l1+2*l2*l4*f5*k3*f3*k1*f1+4*l2*l4*k2*f6*f1*k3*f3+16*l3* l2*f1*k3*f6*f4*k4+16*l3*l2*f1^2*f6^2*k3*k2+8*l2*l4*f0*f6*k2^2*f3+16*l2*l4*f0*f6 *k3*f4*k2-2*l2*l4*f1*k3^2*f5^2+4*l2*l4*f0*f5*k4*k1+8*l2*l4*f0*f5*k1^2*f2-4*l2* l4*k2*k1*f6*f1^2+8*l4*f4*f0*k1*k4*l1-2*l4*f3*k1*f1*k4*l1+16*k2*f6*f1*k3*f4^2*l3 ^2+8*l3^2*f1*k3^2*f6*f4*f5+4*l3*k4*l2*f3*k2*f6*f1+16*l3*l2*f1*f0*f6^2*k2^2-4*l3 *l2*f0*f5^2*f3*k2^2+16*l3*f0^2*f6*k2*f5*k1*l1+4*f0*f5*k1*f3^2*k2*l3*l1-4*l3*l2* f1*f5^2*k3*k4+16*l3*l2*f0*k4*f6*k1*f3+16*f0*k4*f6*k2*l3*l2*f4-2*l3^2*k2*f5^2*f1 *k4+8*l3*f1*f5^2*f0*k2*k1*l1-4*l3^2*f2*f5^2*k3*k4+16*l3^2*f2*f6*k3*f4*k4-16*l3* f0*f4*f1*f6*k2*k1*l1+16*f0*k1*f6*f2*k2*l3*f3*l1-2*k1^2*f5*f1^2*l3*f3*l1+8*f0*f5 *k1^2*f2*l3*f3*l1+8*l3*f1*f5*f0*f6*k2^2*l1-32*l3*f0^2*f6*k1^2*f4*l1+8*l3*f0*f5* f3*k3*k1*f4*l1-16*l3*f0*f5^2*f2*k3*k1*l1+8*l3*f2*k3*f6*f3*k1*f1*l1-16*l3*k3*f6* f4*k1*f1^2*l1+32*l3*f0*f4*f2*f6*k3*k1*l1+32*l3*f0^2*f6^2*k3*k1*l1-32*l3*f2*f0* f6^2*k3^2*l1+16*l3*f1*f0*f6^2*k3*k2*l1+8*f1*k3^2*f6*f4*l3*f3*l1-2*f1*k3^2*f5^2* l3*f3*l1+8*l3*f0*f5*k4*k1*f3*l1-8*l3*l2*f1*f5*f3*f6*k3^2-2*l3*k2*f5^2*f1*k3*l2* f3+8*l3*f0*f5*f3*f6*k3^2*l1+8*l3*k2*f6*f1^2*k3*f5*l1+4*k2*f6*f1*k3*f3^2*l3*l1+8 *f0*f6*k2^2*f3^2*l3*l1+8*f0*k4*f6*k2*l3*f3*l1+8*l3*f1*k3*f6*f3*k4*l1-8*l3*f0*f6 *k3*f3^2*k1*l1+8*l3*f5^2*k3*k1*f1^2*l1+16*f0*f6*k3*f4*k2*l3*f3*l1-16*l3*f0*f5* f2*f6*k3*k2*l1+8*l3*f1*f0*f6*k1^2*f3*l1+16*l3*f5^2*f0^2*k1^2*l1-8*f0*k2*f5^2*k3 *l3*l2*f4+16*l3*l2*f0*f5*f4*f6*k3^2-4*f0*k2*f5^2*k4*l2*l3+16*l3*l2*f0^2*f6*f5* k1^2+16*l3*l2*f2*f0*f6*k1^2*f3+32*l3*l2*f0^2*f6^2*k2*k1+16*l3*l2*f0*f3^2*f6*k2* k1+2*l3*k2*f5*f1*k4*f3*l1-4*l3*k1^2*f1^2*f3*l2*f6+4*l3*k1*f1*f3^2*k3*l2*f6-8*l3 *l2*f0*f5*f3*f6*k3*k2-16*l3*l2*f0*f3*f6^2*k3^2+8*l3*l2*f1*f5*f0*f6*k2*k1+16*l3* l2*f1*f0*f6^2*k3*k1+8*l3*l2*k3*f6*f5*k1*f1^2+16*l3*l2*f2*f1*k3^2*f6^2+16*f1*k3^ 2*f6*f4^2*l3*l2-4*f1*k3^2*f5^2*l3*l2*f4-4*l3*l2*f0*f5^3*k3^2+16*l3*f1^2*k3^2*f6 ^2*l1-4*f0*f5^2*k1*f3*k2*l3^2+16*f0*f6*k2^2*f3*l3*l2*f4+8*k2*f6*f1*k3*f3*l3*l2* f4+32*f0*f6*k3*f4^2*k2*l3*l2-16*l3*l2*f2*f5*f0*f6*k3*k1+16*f0*k1*f6*f3*k3*l3*l2 *f4+16*f2*f0*f6^2*k2^2*l3^2+16*l3^2*f1*f0*f6^2*k2*k1+8*f6*k3*f3*k1*f1*l3^2*f4+ 16*l3^2*f2*k2*f6^2*f1*k3-16*l3^2*f0*k3*f6^2*f3*k2-4*f5^2*k2*f1*k3*l3^2*f4+16*l3 ^2*f0*f5*f4*f6*k3*k2-4*l3^2*f0*f5^3*k3*k2-8*l3^2*k2*f5*f1*k3*f6*f3-2*f5^2*k3*f3 *k1*f1*l3^2-4*f3^2*f0*k1^2*l3^2*f6-8*k1^2*f6*f1^2*l3^2*f4+2*k1^2*f5^2*f1^2*l3^2 +16*f4*f0*f6^2*k3^2*l3^2-4*k3^2*f6*f3^2*l3^2*f4-4*f0*f5^2*k3^2*l3^2*f6-4*f2*f5^ 2*k3^2*l3^2*f4+16*f2*k3^2*f6*f4^2*l3^2-16*l3^2*f1*k3^2*f6^2*f3-32*l3^2*f0*f2*f6 ^2*k3*k1+8*l3^2*f0*f5*k1*f6*f3*k3+16*l3^2*k3*f6^2*k1*f1^2-8*l3^2*f2*k3^2*f6*f3* f5+16*f0*f6*f4^2*k2^2*l3^2+8*l2^2*f0*f6*k2*f3*k4-8*l2*f1*f0*f6*k2*k1*f3*l1-4*l2 ^2*f0*f5^2*k3*k4+16*l2^2*f0*f4*f6*k3*k4+16*l2^2*f2*f0*f6*k1*k4+16*f0*f6*k2*k1* f3*l3^2*f4-4*f0*f5^2*k2^2*l3^2*f4-8*f0*f6*k2^2*f3*l3^2*f5+16*l2*f0^2*f6*k3*f5* k1*l1+16*f3^2*f0*f6*k3*k2*l2*l1-2*k1*f1^2*f3*k2*l2*f5*l1+32*l2*f0*f6*k2*f2^2*k1 *l1+16*l2*f5^2*f0^2*k2*k1*l1+16*l2*f5*f0^2*k1^2*f4*l1+8*l2*f0*f5*f2*k2*k1*f3*l1 -8*l2*f2*k2*f6*f1^2*k1*l1-2*l3^2*f1*f5^3*k3^2-4*l2*f6*k1^2*f1^3*l1-16*l2*f0^2* f6*k1^2*f3*l1+16*l2*f0*f5*k1^2*f2^2*l1-4*l2^2*f6*k1*f1^2*k4+16*l2*f0*f6*k3*f3* k1*f2*l1-4*f0*f5^2*k3^2*l2*f3*l1+4*f0*k2*f5*k4*l2*f3*l1+16*l2*f0*f5*f2*k1*k4*l1 +8*l2*f1*f5^2*f0*k3*k1*l1+32*l2*f0^2*f6^2*k3*k2*l1+8*l2*f1*f5*f0*f6*k3*k2*l1-4* f5*k1^2*f1^2*l2*f2*l1+16*l2*f0*f6*k1^2*f1*f2*l1-8*l2*f1*f5*f0*k1^2*f3*l1+16*l2* f5*f0^2*f6*k2^2*l1+16*f2*f0*f6*k2^2*l2*f3*l1+16*l2^2*f2*f0*f6*k2*k1*f3+16*f2*f0 *f6*k2^2*l2^2*f4-8*f0*f5*f1*k2*k1*f3*l1^2+8*l2*l4*f1*k3^2*f6*f4-4*l4*f5*k2*f1^2 *k1*l1+32*f0*f6*k1^2*f2*l3^2*f4-8*f0*f5^2*k1^2*f2*l3^2+16*f0*f6*k1*k4*l3^2*f4-4 *f0*f5^2*k4*k1*l3^2-4*l3^2*k4*f3^2*k3*f6+8*l3^2*k2*f6*f1*k4*f4: # We now introduce curve whose Jacobian has C5xC5, then specialise # the above defining equation and biquadratic forms, and then get # the two spaces of quintic forms # (mod k4*kummeqn, k3*kummeqn, k2*kummeqn, k1*kummeqn). sex1 := 5*(16 - 12*x + 3*x^2)^2 + (x - 4)^5; sex2 := (16 - 20*x + 5*x^2)^2 + x^5; chk := expand(sex1 - sex2); C := expand( 5*(16 - 12*x + 3*x^2)^2 + (x - 4)^5 ); ff6 := coeff(C,x,6); ff5 := coeff(C,x,5); ff4 := coeff(C,x,4); ff3 := coeff(C,x,3); ff2 := coeff(C,x,2); ff1 := coeff(C,x,1); ff0 := coeff(C,x,0); kummeqn := subs(f6=ff6, f5=ff5, f4=ff4, f3=ff3, f2=ff2, f1=ff1, f0=ff0, kummeqngeneral); # Q1 = T1 = [ (4,G(4)) - infty ] = [ (4, 16*sqrt(5)) - infty] # Q1Q1 = 2*Q1 = { (4, 16*sqrt(5)), (4, 16*sqrt(5)) }. # Q2 = T2 = [ (0,H(0)) - infty ] = [ (0, 16) - infty] # Q2Q2 = 2*Q2 = { (0,16), (0,16) }. # # We now get the Kummer coordinates: Q1k1, Q1k2, Q1k3, Q1k4 # for the point of order 5: Q1 := { (4, 16*sqrt(5)), infty }. Q1k1 := 0; Q1k2 := 1; Q1k3 := 4; Q1k4 := solve( subs( k1=Q1k1, k2 = Q1k2, k3 = Q1k3, kummeqn ) )[1]; chk := simplify( subs( f6=ff6, f5=ff5, f4=ff4, f3=ff3, f2=ff2, f1=ff1, f0=ff0, k1=Q1k1, k2=Q1k2, k3=Q1k3, k4=Q1k4, kummeqn ) ); # We now get the Kummer coordinates: Q1Q1k1, Q1Q1k2, Q1Q1k3, Q1Q1k4 # for the point of order 5: Q1Q1 := { (4, 16*sqrt(5)), (4, 16*sqrt(5)) }. Q1Q1k1 := 1; Q1Q1k2 := 4 + 4; Q1Q1k3 := 4*4; Q1Q1k4 := solve( subs( k1=Q1Q1k1, k2 = Q1Q1k2, k3 = Q1Q1k3, kummeqn ) ); chk := simplify( subs( f6=ff6, f5=ff5, f4=ff4, f3=ff3, f2=ff2, f1=ff1, f0=ff0, k1=Q1Q1k1, k2=Q1Q1k2, k3=Q1Q1k3, k4=Q1Q1k4, kummeqn ) ); # We now get the Kummer coordinates: Q2k1, Q2k2, Q2k3, Q2k4 # for the point of order 5: Q2 := { (0, 16), infty }. Q2k1 := 0; Q2k2 := 1; Q2k3 := 0; Q2k4 := solve( subs( k1=Q2k1, k2 = Q2k2, k3 = Q2k3, kummeqn ) )[1]; chk := simplify( subs( f6=ff6, f5=ff5, f4=ff4, f3=ff3, f2=ff2, f1=ff1, f0=ff0, k1=Q2k1, k2=Q2k2, k3=Q2k3, k4=Q2k4, kummeqn ) ); # We now get the Kummer coordinates: Q2Q2k1, Q2Q2k2, Q2Q2k3, Q2Q2k4 # for the point of order 5: Q2Q2 := { (0, 16), (0,16) }. Q2Q2k1 := 1; Q2Q2k2 := 0 + 0; Q2Q2k3 := 0*0; Q2Q2k4 := solve( subs( k1=Q2Q2k1, k2 = Q2Q2k2, k3 = Q2Q2k3, kummeqn ) ); chk := simplify( subs( f6=ff6, f5=ff5, f4=ff4, f3=ff3, f2=ff2, f1=ff1, f0=ff0, k1=Q2Q2k1, k2=Q2Q2k2, k3=Q2Q2k3, k4=Q2Q2k4, kummeqn ) ); # R_{ij}\bigl( k(P) \bigr) &=& B_{ij}\bigl( k(P), k(Q_1) \bigr), # Rd_{ij}\bigl( k(P) \bigr) &=& B_{ij}\bigl( k(P), k(2*Q_1) \bigr), # S_{ij}\bigl( k(P) \bigr) &=& B_{ij}\bigl( k(P), k(Q_2) \bigr), # Sd_{ij}\bigl( k(P) \bigr) &=& B_{ij}\bigl( k(P), k(2*Q_2) \bigr), R := matrix(4,4); for ii from 1 to 4 do for jj from 1 to 4 do R[ii,jj] := simplify(subs( f6=ff6, f5=ff5, f4=ff4, f3=ff3, f2=ff2, f1=ff1, f0=ff0, l1=Q1k1, l2=Q1k2, l3=Q1k3, l4=Q1k4, BBB[ii,jj] )); od od; Rd := matrix(4,4); for ii from 1 to 4 do for jj from 1 to 4 do Rd[ii,jj] := simplify(subs( f6=ff6, f5=ff5, f4=ff4, f3=ff3, f2=ff2, f1=ff1, f0=ff0, l1=Q1Q1k1, l2=Q1Q1k2, l3=Q1Q1k3, l4=Q1Q1k4, BBB[ii,jj] )); od od; S := matrix(4,4); for ii from 1 to 4 do for jj from 1 to 4 do S[ii,jj] := simplify(subs( f6=ff6, f5=ff5, f4=ff4, f3=ff3, f2=ff2, f1=ff1, f0=ff0, l1=Q2k1, l2=Q2k2, l3=Q2k3, l4=Q2k4, BBB[ii,jj] )); od od; Sd := matrix(4,4); for ii from 1 to 4 do for jj from 1 to 4 do Sd[ii,jj] := simplify(subs( f6=ff6, f5=ff5, f4=ff4, f3=ff3, f2=ff2, f1=ff1, f0=ff0, l1=Q2Q2k1, l2=Q2Q2k2, l3=Q2Q2k3, l4=Q2Q2k4, BBB[ii,jj] )); od od; # kk[i]*R[j,k]*Rd[l,m] + kk[i]*R[j,l]*Rd[k,m] + kk[i]*R[j,m]*Rd[k,l] # + kk[i]*R[k,l]*Rd[j,m] + kk[i]*R[k,m]*Rd[j,l] + kk[i]*R[l,m]*Rd[j,k] # + kk[j]*R[i,k]*Rd[l,m] + kk[j]*R[i,l]*Rd[k,m] + kk[j]*R[i,m]*Rd[k,l] # + kk[j]*R[k,l]*Rd[i,m] + kk[j]*R[k,m]*Rd[i,l] + kk[j]*R[l,m]*Rd[i,k] # + kk[k]*R[j,i]*Rd[l,m] + kk[k]*R[j,l]*Rd[i,m] + kk[k]*R[j,m]*Rd[i,l] # + kk[k]*R[i,l]*Rd[j,m] + kk[k]*R[i,m]*Rd[j,l] + kk[k]*R[l,m]*Rd[j,i] # + kk[l]*R[j,k]*Rd[i,m] + kk[l]*R[j,i]*Rd[k,m] + kk[l]*R[j,m]*Rd[k,i] # + kk[l]*R[k,i]*Rd[j,m] + kk[l]*R[k,m]*Rd[j,i] + kk[l]*R[i,m]*Rd[j,k] # + kk[m]*R[j,k]*Rd[l,i] + kk[m]*R[j,l]*Rd[k,i] + kk[m]*R[j,i]*Rd[k,l] # + kk[m]*R[k,l]*Rd[j,i] + kk[m]*R[k,i]*Rd[j,l] + kk[m]*R[l,i]*Rd[j,k] kk := [k1,k2,k3,k4]; RR := Array(1..4,1..4,1..4,1..4,1..4); for i from 1 to 4 do for j from 1 to 4 do for k from 1 to 4 do for l from 1 to 4 do for m from 1 to 4 do RR[i,j,k,l,m] := expand( kk[i]*R[j,k]*Rd[l,m] + kk[i]*R[j,l]*Rd[k,m] + kk[i]*R[j,m]*Rd[k,l] + kk[i]*R[k,l]*Rd[j,m] + kk[i]*R[k,m]*Rd[j,l] + kk[i]*R[l,m]*Rd[j,k] + kk[j]*R[i,k]*Rd[l,m] + kk[j]*R[i,l]*Rd[k,m] + kk[j]*R[i,m]*Rd[k,l] + kk[j]*R[k,l]*Rd[i,m] + kk[j]*R[k,m]*Rd[i,l] + kk[j]*R[l,m]*Rd[i,k] + kk[k]*R[j,i]*Rd[l,m] + kk[k]*R[j,l]*Rd[i,m] + kk[k]*R[j,m]*Rd[i,l] + kk[k]*R[i,l]*Rd[j,m] + kk[k]*R[i,m]*Rd[j,l] + kk[k]*R[l,m]*Rd[j,i] + kk[l]*R[j,k]*Rd[i,m] + kk[l]*R[j,i]*Rd[k,m] + kk[l]*R[j,m]*Rd[k,i] + kk[l]*R[k,i]*Rd[j,m] + kk[l]*R[k,m]*Rd[j,i] + kk[l]*R[i,m]*Rd[j,k] + kk[m]*R[j,k]*Rd[l,i] + kk[m]*R[j,l]*Rd[k,i] + kk[m]*R[j,i]*Rd[k,l] + kk[m]*R[k,l]*Rd[j,i] + kk[m]*R[k,i]*Rd[j,l] + kk[m]*R[l,i]*Rd[j,k] ) od od od od od; kk := [k1,k2,k3,k4]; SS := Array(1..4,1..4,1..4,1..4,1..4); for i from 1 to 4 do for j from 1 to 4 do for k from 1 to 4 do for l from 1 to 4 do for m from 1 to 4 do SS[i,j,k,l,m] := expand( kk[i]*S[j,k]*Sd[l,m] + kk[i]*S[j,l]*Sd[k,m] + kk[i]*S[j,m]*Sd[k,l] + kk[i]*S[k,l]*Sd[j,m] + kk[i]*S[k,m]*Sd[j,l] + kk[i]*S[l,m]*Sd[j,k] + kk[j]*S[i,k]*Sd[l,m] + kk[j]*S[i,l]*Sd[k,m] + kk[j]*S[i,m]*Sd[k,l] + kk[j]*S[k,l]*Sd[i,m] + kk[j]*S[k,m]*Sd[i,l] + kk[j]*S[l,m]*Sd[i,k] + kk[k]*S[j,i]*Sd[l,m] + kk[k]*S[j,l]*Sd[i,m] + kk[k]*S[j,m]*Sd[i,l] + kk[k]*S[i,l]*Sd[j,m] + kk[k]*S[i,m]*Sd[j,l] + kk[k]*S[l,m]*Sd[j,i] + kk[l]*S[j,k]*Sd[i,m] + kk[l]*S[j,i]*Sd[k,m] + kk[l]*S[j,m]*Sd[k,i] + kk[l]*S[k,i]*Sd[j,m] + kk[l]*S[k,m]*Sd[j,i] + kk[l]*S[i,m]*Sd[j,k] + kk[m]*S[j,k]*Sd[l,i] + kk[m]*S[j,l]*Sd[k,i] + kk[m]*S[j,i]*Sd[k,l] + kk[m]*S[k,l]*Sd[j,i] + kk[m]*S[k,i]*Sd[j,l] + kk[m]*S[l,i]*Sd[j,k] ) od od od od od; # Now adjust: (mod k4*kummeqn, k3*kummeqn, k2*kummeqn, k1*kummeqn). # For each quintic, remove the k2^2*k4^2*k4 term, and then # the k2^2*k4^2*k3, k2^2*k4^2*k2, k2^2*k4^2*k1 terms. for i from 1 to 4 do for j from 1 to 4 do for k from 1 to 4 do for l from 1 to 4 do for m from 1 to 4 do RR[i,j,k,l,m] := expand(RR[i,j,k,l,m]-coeff(coeff(RR[i,j,k,l,m],k2,2),k4,3)*k4*kummeqn) od od od od od; for i from 1 to 4 do for j from 1 to 4 do for k from 1 to 4 do for l from 1 to 4 do for m from 1 to 4 do RR[i,j,k,l,m] := expand(RR[i,j,k,l,m] -coeff(coeff(coeff(RR[i,j,k,l,m],k2,2),k4,2),k3,1)*k3*kummeqn) od od od od od; for i from 1 to 4 do for j from 1 to 4 do for k from 1 to 4 do for l from 1 to 4 do for m from 1 to 4 do RR[i,j,k,l,m] := expand(RR[i,j,k,l,m]-coeff(coeff(RR[i,j,k,l,m],k2,3),k4,2)*k2*kummeqn) od od od od od; for i from 1 to 4 do for j from 1 to 4 do for k from 1 to 4 do for l from 1 to 4 do for m from 1 to 4 do RR[i,j,k,l,m] := expand(RR[i,j,k,l,m] -coeff(coeff(coeff(RR[i,j,k,l,m],k2,2),k4,2),k1,1)*k1*kummeqn) od od od od od; for i from 1 to 4 do for j from 1 to 4 do for k from 1 to 4 do for l from 1 to 4 do for m from 1 to 4 do SS[i,j,k,l,m] := expand(SS[i,j,k,l,m]-coeff(coeff(SS[i,j,k,l,m],k2,2),k4,3)*k4*kummeqn) od od od od od; for i from 1 to 4 do for j from 1 to 4 do for k from 1 to 4 do for l from 1 to 4 do for m from 1 to 4 do SS[i,j,k,l,m] := expand(SS[i,j,k,l,m] -coeff(coeff(coeff(SS[i,j,k,l,m],k2,2),k4,2),k3,1)*k3*kummeqn) od od od od od; for i from 1 to 4 do for j from 1 to 4 do for k from 1 to 4 do for l from 1 to 4 do for m from 1 to 4 do SS[i,j,k,l,m] := expand(SS[i,j,k,l,m]-coeff(coeff(SS[i,j,k,l,m],k2,3),k4,2)*k2*kummeqn) od od od od od; for i from 1 to 4 do for j from 1 to 4 do for k from 1 to 4 do for l from 1 to 4 do for m from 1 to 4 do SS[i,j,k,l,m] := expand(SS[i,j,k,l,m] -coeff(coeff(coeff(SS[i,j,k,l,m],k2,2),k4,2),k1,1)*k1*kummeqn) od od od od od; # Step 1. Get the 4 quintic forms l1,l2,l3,l4 of the map, diag as # 1*k1*k4^4 + 0*k2*k4^4 + 0*k3*k4^4 + 0*k4^5, # 0*k1*k4^4 + 1*k2*k4^4 + 0*k3*k4^4 + 0*k4^5, # 0*k1*k4^4 + 0*k2*k4^4 + 1*k3*k4^4 + 0*k4^5, # 0*k1*k4^4 + 0*k2*k4^4 + 0*k3*k4^4 + 1*k4^5. # Step 2. Find eq satisfied by l1,l2,l3,l4. # (if l1,l2,l3 already correct, can map irrat pts of order 3 on C # to find l4). # Step 3. Find the linear change in the l's (probably just adjust l4), # so that the qn is transparently a Kummer eqn. # Step 4. Get the correct quadratic twist, using a rank 1 example. getarr := proc(quinticform); # This procedure inputs a quintic in k1,k2,k3,k4 and gets array of coeffs. [ coeff(coeff(coeff(coeff(quinticform,k4,5),k3,0),k2,0),k1,0), coeff(coeff(coeff(coeff(quinticform,k4,4),k3,1),k2,0),k1,0), coeff(coeff(coeff(coeff(quinticform,k4,4),k3,0),k2,1),k1,0), coeff(coeff(coeff(coeff(quinticform,k4,4),k3,0),k2,0),k1,1), coeff(coeff(coeff(coeff(quinticform,k4,3),k3,2),k2,0),k1,0), coeff(coeff(coeff(coeff(quinticform,k4,3),k3,1),k2,1),k1,0), coeff(coeff(coeff(coeff(quinticform,k4,3),k3,1),k2,0),k1,1), coeff(coeff(coeff(coeff(quinticform,k4,3),k3,0),k2,2),k1,0), coeff(coeff(coeff(coeff(quinticform,k4,3),k3,0),k2,1),k1,1), coeff(coeff(coeff(coeff(quinticform,k4,3),k3,0),k2,0),k1,2), coeff(coeff(coeff(coeff(quinticform,k4,2),k3,3),k2,0),k1,0), coeff(coeff(coeff(coeff(quinticform,k4,2),k3,2),k2,1),k1,0), coeff(coeff(coeff(coeff(quinticform,k4,2),k3,2),k2,0),k1,1), coeff(coeff(coeff(coeff(quinticform,k4,2),k3,1),k2,2),k1,0), coeff(coeff(coeff(coeff(quinticform,k4,2),k3,1),k2,1),k1,1), coeff(coeff(coeff(coeff(quinticform,k4,2),k3,1),k2,0),k1,2), coeff(coeff(coeff(coeff(quinticform,k4,2),k3,0),k2,3),k1,0), coeff(coeff(coeff(coeff(quinticform,k4,2),k3,0),k2,2),k1,1), coeff(coeff(coeff(coeff(quinticform,k4,2),k3,0),k2,1),k1,2), coeff(coeff(coeff(coeff(quinticform,k4,2),k3,0),k2,0),k1,3), coeff(coeff(coeff(coeff(quinticform,k4,1),k3,4),k2,0),k1,0), coeff(coeff(coeff(coeff(quinticform,k4,1),k3,3),k2,1),k1,0), coeff(coeff(coeff(coeff(quinticform,k4,1),k3,3),k2,0),k1,1), coeff(coeff(coeff(coeff(quinticform,k4,1),k3,2),k2,2),k1,0), coeff(coeff(coeff(coeff(quinticform,k4,1),k3,2),k2,1),k1,1), coeff(coeff(coeff(coeff(quinticform,k4,1),k3,2),k2,0),k1,2), coeff(coeff(coeff(coeff(quinticform,k4,1),k3,1),k2,3),k1,0), coeff(coeff(coeff(coeff(quinticform,k4,1),k3,1),k2,2),k1,1), coeff(coeff(coeff(coeff(quinticform,k4,1),k3,1),k2,1),k1,2), coeff(coeff(coeff(coeff(quinticform,k4,1),k3,1),k2,0),k1,3), coeff(coeff(coeff(coeff(quinticform,k4,1),k3,0),k2,4),k1,0), coeff(coeff(coeff(coeff(quinticform,k4,1),k3,0),k2,3),k1,1), coeff(coeff(coeff(coeff(quinticform,k4,1),k3,0),k2,2),k1,2), coeff(coeff(coeff(coeff(quinticform,k4,1),k3,0),k2,1),k1,3), coeff(coeff(coeff(coeff(quinticform,k4,1),k3,0),k2,0),k1,4), coeff(coeff(coeff(coeff(quinticform,k4,0),k3,5),k2,0),k1,0), coeff(coeff(coeff(coeff(quinticform,k4,0),k3,4),k2,1),k1,0), coeff(coeff(coeff(coeff(quinticform,k4,0),k3,4),k2,0),k1,1), coeff(coeff(coeff(coeff(quinticform,k4,0),k3,3),k2,2),k1,0), coeff(coeff(coeff(coeff(quinticform,k4,0),k3,3),k2,1),k1,1), coeff(coeff(coeff(coeff(quinticform,k4,0),k3,3),k2,0),k1,2), coeff(coeff(coeff(coeff(quinticform,k4,0),k3,2),k2,3),k1,0), coeff(coeff(coeff(coeff(quinticform,k4,0),k3,2),k2,2),k1,1), coeff(coeff(coeff(coeff(quinticform,k4,0),k3,2),k2,1),k1,2), coeff(coeff(coeff(coeff(quinticform,k4,0),k3,2),k2,0),k1,3), coeff(coeff(coeff(coeff(quinticform,k4,0),k3,1),k2,4),k1,0), coeff(coeff(coeff(coeff(quinticform,k4,0),k3,1),k2,3),k1,1), coeff(coeff(coeff(coeff(quinticform,k4,0),k3,1),k2,2),k1,2), coeff(coeff(coeff(coeff(quinticform,k4,0),k3,1),k2,1),k1,3), coeff(coeff(coeff(coeff(quinticform,k4,0),k3,1),k2,0),k1,4), coeff(coeff(coeff(coeff(quinticform,k4,0),k3,0),k2,5),k1,0), coeff(coeff(coeff(coeff(quinticform,k4,0),k3,0),k2,4),k1,1), coeff(coeff(coeff(coeff(quinticform,k4,0),k3,0),k2,3),k1,2), coeff(coeff(coeff(coeff(quinticform,k4,0),k3,0),k2,2),k1,3), coeff(coeff(coeff(coeff(quinticform,k4,0),k3,0),k2,1),k1,4), coeff(coeff(coeff(coeff(quinticform,k4,0),k3,0),k2,0),k1,5) ]; end; # The following puts the 1024 Q1-cubics as 1024 rows of coeffs (1024x56 matrix). Q1arr := []; for i from 1 to 4 do for j from 1 to 4 do for k from 1 to 4 do for l from 1 to 4 do for m from 1 to 4 do Q1arr := [ op(Q1arr), getarr(RR[i,j,k,l,m]) ]; od od od od od; Q1mtx := matrix( Q1arr ); Q1mtx := gausselim(Q1mtx); # The following gives: 12, so that this is a 12-dimensional rowspace. rank(Q1mtx); # The following puts the 1024 Q2-cubics as 1024 rows of coeffs (1024x56 matrix). Q2arr := []; for i from 1 to 4 do for j from 1 to 4 do for k from 1 to 4 do for l from 1 to 4 do for m from 1 to 4 do Q2arr := [ op(Q2arr), getarr(SS[i,j,k,l,m]) ]; od od od od od; Q2mtx := matrix( Q2arr ); Q2mtx := gausselim(Q2mtx); # The following gives: 12, so that this is a 12-dimensional rowspace. rank(Q2mtx); # We see that Q1mtx, Q2mtx each have rowspace of dimension 12. # We shall now find a basis for the intersections of these rowspaces # by first finding nullQ1,nullQ2, whose rows are the transposes of # the bases for the nullspaces of Q1mtx,Q2mtx, then find nullQ1Q2, # the matrix obtained by placing the rows of nullQ1 above the rows # of nullQ2, and then Q1intQ2, whose rows are the transposes of # the basis elts of the nullspace of nullQ1Q2. QQ1 := nullspace(Q1mtx); nullQ1 := []; for i from 1 to nops(QQ1) do nullQ1 := [ op(nullQ1), convert(QQ1[i],list) ] od; nullQ1 := matrix(nullQ1); QQ2 := nullspace(Q2mtx); nullQ2 := []; for i from 1 to nops(QQ2) do nullQ2 := [ op(nullQ2), convert(QQ2[i],list) ] od; nullQ2 := matrix(nullQ2); nullQ1Q2 := matrix([op(convert(nullQ1,listlist)), op(convert(nullQ2,listlist))]); # The following find a matrix whose 4 rows are a basis for the # intersection of the rowspace of Q1mtx and the rowspace of Q2mtx. nullnullQ1Q2 := nullspace( nullQ1Q2 ); Q1intQ2 := []; for i from 1 to nops(nullnullQ1Q2) do Q1intQ2 := [ op(Q1intQ2), convert(nullnullQ1Q2[i],list) ] od; Q1intQ2 := matrix( Q1intQ2 ); # So far, to here. getquinticform := proc(arrr) arrr[1]*k4^5 + arrr[2]*k4^4*k3^1 + arrr[3]*k4^4*k2^1 + arrr[4]*k4^4*k1^1 + arrr[5]*k4^3*k3^2 + arrr[6]*k4^3*k3^1*k2^1 + arrr[7]*k4^3*k3^1*k1^1 + arrr[8]*k4^3*k2^2 + arrr[9]*k4^3*k2^1*k1^1 + arrr[10]*k4^3*k1^2 + arrr[11]*k4^2*k3^3 + arrr[12]*k4^2*k3^2*k2^1 + arrr[13]*k4^2*k3^2*k1^1 + arrr[14]*k4^2*k3^1*k2^2 + arrr[15]*k4^2*k3^1*k2^1*k1^1 + arrr[16]*k4^2*k3^1*k1^2 + arrr[17]*k4^2*k2^3 + arrr[18]*k4^2*k2^2*k1^1 + arrr[19]*k4^2*k2^1*k1^2 + arrr[20]*k4^2*k1^3 + arrr[21]*k4^1*k3^4 + arrr[22]*k4^1*k3^3*k2^1 + arrr[23]*k4^1*k3^3*k1^1 + arrr[24]*k4^1*k3^2*k2^2 + arrr[25]*k4^1*k3^2*k2^1*k1^1 + arrr[26]*k4^1*k3^2*k1^2 + arrr[27]*k4^1*k3^1*k2^3 + arrr[28]*k4^1*k3^1*k2^2*k1^1 + arrr[29]*k4^1*k3^1*k2^1*k1^2 + arrr[30]*k4^1*k3^1*k1^3 + arrr[31]*k4^1*k2^4 + arrr[32]*k4^1*k2^3*k1^1 + arrr[33]*k4^1*k2^2*k1^2 + arrr[34]*k4^1*k2^1*k1^3 + arrr[35]*k4^1*k1^4 + arrr[36]*k3^5 + arrr[37]*k3^4*k2^1 + arrr[38]*k3^4*k1^1 + arrr[39]*k3^3*k2^2 + arrr[40]*k3^3*k2^1*k1^1 + arrr[41]*k3^3*k1^2 + arrr[42]*k3^2*k2^3 + arrr[43]*k3^2*k2^2*k1^1 + arrr[44]*k3^2*k2^1*k1^2 + arrr[45]*k3^2*k1^3 + arrr[46]*k3^1*k2^4 + arrr[47]*k3^1*k2^3*k1^1 + arrr[48]*k3^1*k2^2*k1^2 + arrr[49]*k3^1*k2^1*k1^3 + arrr[50]*k3^1*k1^4 + arrr[51]*k2^5 + arrr[52]*k2^4*k1^1 + arrr[53]*k2^3*k1^2 + arrr[54]*k2^2*k1^3 + arrr[55]*k2^1*k1^4 + arrr[56]*k1^5 end; r1 := getquinticform( convert(Q1intQ2, listlist)[1] ); r2 := getquinticform( convert(Q1intQ2, listlist)[2] ); r3 := getquinticform( convert(Q1intQ2, listlist)[3] ); r4 := getquinticform( convert(Q1intQ2, listlist)[4] ); m := matrix( [ [ coeff(coeff(r1,k1,1),k4,4), coeff(coeff(r1,k2,1),k4,4), coeff(coeff(r1,k3,1),k4,4), coeff(r1,k4,5) ], [ coeff(coeff(r2,k1,1),k4,4), coeff(coeff(r2,k2,1),k4,4), coeff(coeff(r2,k3,1),k4,4), coeff(r2,k4,5) ], [ coeff(coeff(r3,k1,1),k4,4), coeff(coeff(r3,k2,1),k4,4), coeff(coeff(r3,k3,1),k4,4), coeff(r3,k4,5) ], [ coeff(coeff(r4,k1,1),k4,4), coeff(coeff(r4,k2,1),k4,4), coeff(coeff(r4,k3,1),k4,4), coeff(r4,k4,5) ] ] ); lmtx := multiply( inverse(m), matrix([ [r1], [r2], [r3], [r4] ]) ); # The following is the diagonalised quintic map. l1 := lmtx[1,1]; l2 := lmtx[2,1]; l3 := lmtx[3,1]; l4 := lmtx[4,1]; # The following check that Q1, 2*Q1, Q2, 2*Q2 are indeed in the kernel # of the quintic map (k1,k2,k3,k4) |--> (l1,l2,l3,l4). subs(k1=Q1k1,k2=Q1k2,k3=Q1k3,k4=Q1k4, l1); subs(k1=Q1k1,k2=Q1k2,k3=Q1k3,k4=Q1k4, l2); subs(k1=Q1k1,k2=Q1k2,k3=Q1k3,k4=Q1k4, l3); subs(k1=Q1k1,k2=Q1k2,k3=Q1k3,k4=Q1k4, l4); subs(k1=Q1Q1k1,k2=Q1Q1k2,k3=Q1Q1k3,k4=Q1Q1k4, l1); subs(k1=Q1Q1k1,k2=Q1Q1k2,k3=Q1Q1k3,k4=Q1Q1k4, l2); subs(k1=Q1Q1k1,k2=Q1Q1k2,k3=Q1Q1k3,k4=Q1Q1k4, l3); subs(k1=Q1Q1k1,k2=Q1Q1k2,k3=Q1Q1k3,k4=Q1Q1k4, l4); subs(k1=Q2k1,k2=Q2k2,k3=Q2k3,k4=Q2k4, l1); subs(k1=Q2k1,k2=Q2k2,k3=Q2k3,k4=Q2k4, l2); subs(k1=Q2k1,k2=Q2k2,k3=Q2k3,k4=Q2k4, l3); subs(k1=Q2k1,k2=Q2k2,k3=Q2k3,k4=Q2k4, l4); subs(k1=Q2Q2k1,k2=Q2Q2k2,k3=Q2Q2k3,k4=Q2Q2k4, l1); subs(k1=Q2Q2k1,k2=Q2Q2k2,k3=Q2Q2k3,k4=Q2Q2k4, l2); subs(k1=Q2Q2k1,k2=Q2Q2k2,k3=Q2Q2k3,k4=Q2Q2k4, l3); subs(k1=Q2Q2k1,k2=Q2Q2k2,k3=Q2Q2k3,k4=Q2Q2k4, l4); # # Now try to find the quartic satsfied by l1,l2,l3,l4. # We hope that this quartic is: (l2^2-4*l1*l3)*l4^2 + terms deg <= 1 in l4. # Subtracting k4^16*kummeqn removes k4^18, the highest degree term in k4. part1 := expand( (l2^2-4*l1*l3)*l4^2 - k4^16*kummeqn); # Now extract the k4^17 coefficient of part1. part1k4to17 := coeff(part1,k4,17); subs( k1=ll1, k2=ll2, k3=ll3, k4=ll4, part1k4to17 ); # -104*ll2^2*ll3+68*ll2*ll3^2-6976*ll1^2*ll3+384*ll1*ll2^2-444*ll1*ll3^2 # +3328*ll1^2*ll2+1360*ll1*ll2*ll3-13*ll3^3-3072*ll1^3 part2 := expand( part1 - ( -104*l2^2*l3+68*l2*l3^2-6976*l1^2*l3+384*l1*l2^2-444*l1*l3^2 +3328*l1^2*l2+1360*l1*l2*l3-13*l3^3-3072*l1^3 )*l4 ); # Now extract the k4^16 coefficient of part2. part2k4to16 := coeff(part2,k4,16); subs( k1=ll1, k2=ll2, k3=ll3, k4=ll4, part2k4to16 ); # 13424*ll2^2*ll3^2-10568*ll2*ll3^3+10816*ll2^3*ll3+2648960*ll1^2*ll3^2 # +174080*ll1*ll2^3+121296*ll1*ll3^3-8609792*ll1^3*ll2+16238592*ll1^3*ll3 # +66560*ll1^2*ll2^2+291840*ll1*ll2^2*ll3-521088*ll1*ll2*ll3^2+1691*ll3^4 # +7110656*ll1^4-5587456*ll1^2*ll2*ll3 part3 := expand( part2 - ( 13424*l2^2*l3^2-10568*l2*l3^3+10816*l2^3*l3+2648960*l1^2*l3^2 +174080*l1*l2^3+121296*l1*l3^3-8609792*l1^3*l2+16238592*l1^3*l3 +66560*l1^2*l2^2+291840*l1*l2^2*l3-521088*l1*l2*l3^2+1691*l3^4 +7110656*l1^4-5587456*l1^2*l2*l3 ) ); # The above is not divisible by kummeqn, so we have not yet found # the quartic satisfied by l1,l2,l3,l4. Now try using local power series. # # Recall the usual embedding (a0,..,a15) of the Jacobian variety at: # http://people.maths.ox.ac.uk/flynn/genus2/jacobian.variety/defining.equations # # The following 72 quadratic forms eqn(1),..,eqn(72) give a set # of defining equations for the jacobian variety of the # curve of genus 2: # Y**2 = f6*X**6 + f5*X**5 + f4*X**4 + f3*X**3 + f2*X**2 + f1*X + f0 # using the embedding into projective 15-space give by # the following functions on the divisor # {(x,y),(u,v)} = (x,y) + (u,v) - infty+ - infty-. # Note that a0,a3,a4,a5,a10,a11,a12,a13,a14,a15 are even # and a1,a2,a6,a7,a8,a9 are odd. # The defining equations have been organised so that # eqn(1)..eqn(21) are purely even*even terms # eqn(22)..eqn(42) have odd*odd and even*even terms # and eqn(43)..eqn(72) have purely odd*even terms. # # a15 := (x-u)**2; a14 := 1; a13 := x + u; a12 := x*u; # a11 := x*u*(x+u); a10 := (x*u)**2; a9 := (y-v)/(x-u); # a8 := (u*y-x*v)/(x-u); a7 := (u**2*y-x**2*v)/(x-u); # a6 := (u**3*y-x**3*v)/(x-u); a5 := (f0xu-2*y*v)/((x-u)**2); # a4 := (f1xu-(x+u)*y*v)/((x-u)**2); a3 := (x*u)*a5; # ## where f0xu, f1xu are # f0xu := 2*f0+f1*(x+u)+2*f2*(x*u)+f3*(x+u)*(x*u) # +2*f4*(x*u)**2+f5*(x+u)*(x*u)**2+2*f6*(x*u)**3; # f1xu := f0*(x+u)+2*f1*(x*u)+f2*(x+u)*(x*u)+2*f3*(x*u)**2 # +f4*(x+u)*(x*u)**2+2*f5*(x*u)**3+f6*(x+u)*(x*u)**3; # a2:=(gxu*y-gux*v)/((x-u)**3); a1:=(hxu*y-hux*v)/((x-u)**3); # where gxu, gux, hxu, hux are # gxu := f0*4+f1*(x+3*u)+f2*(2*x*u+2*u**2)+f3*(3*x*u**2+u**3) # +f4*(4*x*u**3)+f5*x*(x*u**3+3*u**4)+f6*2*x*(x*u**4+u**5); # gux := f0*4+f1*(u+3*x)+f2*(2*u*x+2*x**2)+f3*(3*u*x**2+x**3) # +f4*(4*u*x**3)+f5*u*(u*x**3+3*x**4)+f6*2*u*(u*x**4+x**5); # hxu := f0*2*(x+u)+f1*u*(3*x+u)+f2*4*x*u**2+f3*x*u**2*(x+3*u) # +f4*2*x*u**3*(x+u)+f5*x*u**4*(3*x+u)+f6*4*x**2*u**5; # hux := f0*2*(u+x)+f1*x*(3*u+x)+f2*4*u*x**2+f3*u*x**2*(u+3*x) # +f4*2*u*x**3*(u+x)+f5*u*x**4*(3*u+x)+f6*4*u**2*x**5; # a0 :=a5**2; # # We shall only requre here three of these. eqn(8) := -f1*a14**2-f3*a14*a12+2*f4*a13*a12-f5*a12**2-2*a4*a14-2*f4* a14*a11+a5*a13; eqn(17) := -a10*a14+a12**2; eqn(20) := -a12*a13+a11*a14; # The following removes the 12^2 and a12*a13 terms from eqn(8): eqnvar := expand( eqn(8) + f5*eqn(17) + 2*f4*eqn(20) ); a5a13 := expand( a5*a13 - eqnvar ); a5x1plusx2 := simplify(a5a13/a14); # This gives a5x1plusx2 (which is: a5*(x1+x2)) as: # a5x1plusx2 := f1*a14+f3*a12+2*a4+f5*a10; # The following is just by the definition of a3: a5x1timesx2 := a3; a5a5 := a0; # Then (k1,k2,k3,k4) = (a5, a5x1plusx2, a5x1timesx2, a5a5). # = (a5, f1*a14+f3*a12+2*a4+f5*a10, a3, a0). # As usual, let si = ai/a0, for i=0..15. # Then (k1,k2,k3,k4) = (s5, f1*s14+f3*s12+2*s4+f5*s10, s3, s0). # We can express each si as a power series in the local parameters s1,s2. # # The following is from the file which gives the local power series. # The following file gives the local power series expansions # of the local coordinates: s0,..,s15 in terms of the local # parameters s1,s2 up to terms of degree 3 in f0,..,f6. # (where si=ai/a0 for all i, where a0,..,a15 are as defined # in the file genus2/jacobian.variety/defining.equations) # This means that: # s3,s4,s5 are given up to (and including) degree 8 in s1,s2, # s6,s7,s8,s9 are given up to (and including) degree 9 in s1,s2, # s10,...,s15 are given up to (and including) degree 10 in s1,s2. # Note that s0,s1,s2 are given exactly, and that the expansions # of s3,s4,s5 contain terms only of even degree in s1,s2, # s6,s7,s8,s9 contain terms only of odd degree in s1,s2, # and s10,..,s15 contain terms only of even degree in s1,s2. # As a consequence, every power series given in this file # is o.k. up to and including degree 9 in s1,s2. s0 := 1; s1 := s1; s2 := s2; s3 := s1^2-f0*s2^4-f4*s1^4-5*f4^3*s1^8+2*f4^2*s1^6-f2*f5^2*s1^8-5*f2^2*f0*s2^8- f6^2*f0*s1^8-f6*f3^2*s1^8-f2*f5*f1*s1^4*s2^4-14*f2*f6*f0*s1^4*s2^4-16*f2*f6*f4* s1^8-4*f2*f5*f0*s1^3*s2^5-4*f2*f0*f4*s1^2*s2^6+3*f6*f5*f1*s1^8+3*f6*f1^2*s1^4* s2^4+f5*f1*s1^4*s2^2-f5*f3*s1^6+12*f6*f5*f0*s1^7*s2-8*f6*f1*f0*s1^3*s2^5-24*f6* f1*f4*s1^7*s2+6*f6*f1*f3*s1^6*s2^2-52*f6*f0^2*s1^2*s2^6-40*f6*f0*f4*s1^6*s2^2+ 24*f6*f0*f3*s1^5*s2^3-2*f5^2*f1*s1^7*s2-2*f5^2*f0*s1^6*s2^2-2*f5*f1*f0*s1^2*s2^ 6-4*f5*f1*f4*s1^6*s2^2-20*f5*f0^2*s1*s2^7-12*f5*f0*f4*s1^5*s2^3+5*f5*f0*f3*s1^4 *s2^4+5*f5*f4*f3*s1^8+2*f1*f0*f3*s2^8-9*f0^2*f4*s2^8-6*f0*f4^2*s1^4*s2^4+4*f2* f6*s1^6+2*f2*f0*s2^6+8*f6*f1*s1^5*s2+18*f6*f0*s1^4*s2^2+4*f5*f0*s1^3*s2^3+2*f0* f4*s1^2*s2^4; s4 := s1*s2-f2^2*f5*s1^4*s2^4+f2*f6*f5*s1^8-f6^2*f1*s1^8+f2*f6*f1*s1^4*s2^4+f5^ 2*f1*s1^6*s2^2-f5*f0^2*s2^8+f5*f1^2*s1^2*s2^6+f6*f3*s1^6+f1*f4*s1^2*s2^4+f0*f3* s2^6+f1*f0*f4*s2^8+f5*f0*f4*s1^4*s2^4-f1*f4^2*s1^4*s2^4+f2*f5*s1^4*s2^2-4*f2*f6 *f0*s1^3*s2^5-8*f2*f6*f4*s1^7*s2+2*f2*f6*f3*s1^6*s2^2-3*f2*f5*f1*s1^3*s2^5-10* f2*f5*f0*s1^2*s2^6-2*f2*f5*f4*s1^6*s2^2-2*f2*f1*f4*s1^2*s2^6-8*f2*f0*f4*s1*s2^7 -3*f2*f0*f3*s2^8-4*f6^2*f0*s1^7*s2+5*f6*f5*f1*s1^7*s2+16*f6*f5*f0*s1^6*s2^2+6* f6*f1^2*s1^3*s2^5+16*f6*f1*f0*s1^2*s2^6-10*f6*f1*f4*s1^6*s2^2+10*f6*f1*f3*s1^5* s2^3-4*f6*f0^2*s1*s2^7-4*f6*f0*f4*s1^5*s2^3+30*f6*f0*f3*s1^4*s2^4-3*f6*f4*f3*s1 ^8+6*f5^2*f0*s1^5*s2^3+5*f5*f1*f0*s1*s2^7-3*f5*f1*f4*s1^5*s2^3+2*f5*f1*f3*s1^4* s2^4+10*f5*f0*f3*s1^3*s2^5+2*f0*f4*f3*s1^2*s2^6+4*f2*f6*s1^5*s2+9*f6*f1*s1^4*s2 ^2+20*f6*f0*s1^3*s2^3+3*f5*f1*s1^3*s2^3+9*f5*f0*s1^2*s2^4+4*f0*f4*s1*s2^5; s5 := s2^2-f2*s2^4-f6*s1^4-5*f2^3*s2^8+2*f2^2*s2^6-f6*f0^2*s2^8-f0*f3^2*s2^8-f1 ^2*f4*s2^8-f5*f1*f4*s1^4*s2^4-6*f2^2*f6*s1^4*s2^4-9*f2*f6^2*s1^8-12*f2*f6*f1*s1 ^3*s2^5+f5*f1*s1^2*s2^4-f1*f3*s2^6-40*f2*f6*f0*s1^2*s2^6-4*f2*f6*f4*s1^6*s2^2-4 *f2*f5*f1*s1^2*s2^6-24*f2*f5*f0*s1*s2^7+5*f2*f1*f3*s2^8-16*f2*f0*f4*s2^8-20*f6^ 2*f1*s1^7*s2-52*f6^2*f0*s1^6*s2^2-2*f6*f5*f1*s1^6*s2^2-8*f6*f5*f0*s1^5*s2^3+2* f6*f5*f3*s1^8-2*f6*f1^2*s1^2*s2^6+12*f6*f1*f0*s1*s2^7-4*f6*f1*f4*s1^5*s2^3+5*f6 *f1*f3*s1^4*s2^4-14*f6*f0*f4*s1^4*s2^4+24*f6*f0*f3*s1^3*s2^5-5*f6*f4^2*s1^8+3* f5^2*f0*s1^4*s2^4-2*f5*f1^2*s1*s2^7+3*f5*f1*f0*s2^8+6*f5*f0*f3*s1^2*s2^6+2*f2* f6*s1^4*s2^2+4*f6*f1*s1^3*s2^3+18*f6*f0*s1^2*s2^4+2*f6*f4*s1^6+8*f5*f0*s1*s2^5+ 4*f0*f4*s2^6; s6 := s1^3-f4*s1^5-5*f4^3*s1^9+2*f4^2*s1^7-f2*f5^2*s1^9-f6*f3^2*s1^9-f6^2*f0*s1 ^9-f2*f1*f4*s1^4*s2^5+f0^2*f3*s2^9-f5*f3*s1^7-f1*f0*f3*s1*s2^8+f3*s1^4*s2+4*f2^ 3*s1^3*s2^6+20*f2^2*f6*s1^7*s2^2+6*f2^2*f1*s1^2*s2^7+9*f2^2*f0*s1*s2^8+2*f2^2* f4*s1^5*s2^4+70*f2*f6*f1*s1^6*s2^3+120*f2*f6*f0*s1^5*s2^4-22*f2*f6*f4*s1^9+12* f2*f6*f3*s1^8*s2+7*f2*f5*f1*s1^5*s2^4+f1^2*s1*s2^6+f1*f0*s2^7+f1*f4*s1^4*s2^3+ 24*f2*f5*f0*s1^4*s2^5-6*f2*f5*f4*s1^8*s2-2*f2*f5*f3*s1^7*s2^2-3*f2*f1^2*s1*s2^8 -3*f2*f1*f0*s2^9-8*f2*f1*f3*s1^3*s2^6-12*f2*f0*f3*s1^2*s2^7+4*f2*f4^2*s1^7*s2^2 +5*f6*f5*f1*s1^9+22*f6*f5*f0*s1^8*s2+66*f6*f1^2*s1^5*s2^4+231*f6*f1*f0*s1^4*s2^ 5-35*f6*f1*f4*s1^8*s2+34*f6*f1*f3*s1^7*s2^2+148*f6*f0^2*s1^3*s2^6-56*f6*f0*f4* s1^7*s2^2+96*f6*f0*f3*s1^6*s2^3+8*f5^2*f0*s1^7*s2^2+19*f5*f1^2*s1^4*s2^5+90*f5* f1*f0*s1^3*s2^6-16*f5*f1*f4*s1^7*s2^2+6*f5*f1*f3*s1^6*s2^3+64*f5*f0^2*s1^2*s2^7 -24*f5*f0*f4*s1^6*s2^3+39*f5*f0*f3*s1^5*s2^4+5*f5*f4*f3*s1^9-2*f5*f3^2*s1^8*s2+ 4*f1^2*f4*s1^3*s2^6-3*f1^2*f3*s1^2*s2^7+38*f1*f0*f4*s1^2*s2^7-2*f1*f4^2*s1^6*s2 ^3-3*f1*f4*f3*s1^5*s2^4+31*f0^2*f4*s1*s2^8-14*f0*f4^2*s1^5*s2^4+10*f0*f4*f3*s1^ 4*s2^5+4*f0*f3^2*s1^3*s2^6+5*f4^2*f3*s1^8*s2-2*f2^2*s1^3*s2^4+6*f2*f6*s1^7+2*f2 *f5*s1^6*s2-3*f2*f1*s1^2*s2^5-4*f2*f0*s1*s2^6-2*f2*f4*s1^5*s2^2+13*f6*f1*s1^6* s2+30*f6*f0*s1^5*s2^2+7*f5*f1*s1^5*s2^2+18*f5*f0*s1^4*s2^3+3*f1*f3*s1^3*s2^4+10 *f0*f4*s1^3*s2^4+6*f0*f3*s1^2*s2^5-2*f4*f3*s1^6*s2+2*f2*s1^3*s2^2+3*f1*s1^2*s2^ 3+3*f0*s1*s2^4; s7 := s1^2*s2+f0*s2^5-f6^2*f1*s1^9-f2*f5^2*s1^8*s2-f6*f3^2*s1^8*s2-f5*f3*s1^6* s2+5*f2^2*f0*s2^9+f1*s1*s2^4-f4*s1^4*s2+5*f2^2*f1*s1*s2^8+8*f2*f6*f1*s1^5*s2^4+ 6*f2*f6*f0*s1^4*s2^5-16*f2*f6*f4*s1^8*s2-2*f2*f5*f1*s1^4*s2^5-8*f2*f5*f0*s1^3* s2^6-12*f2*f0*f4*s1^2*s2^7-6*f2*f0*f3*s1*s2^8-7*f6^2*f0*s1^8*s2+2*f6*f5*f1*s1^8 *s2+8*f6*f5*f0*s1^7*s2^2+17*f6*f1^2*s1^4*s2^5+76*f6*f1*f0*s1^3*s2^6-24*f6*f1*f4 *s1^7*s2^2+8*f6*f1*f3*s1^6*s2^3+44*f6*f0^2*s1^2*s2^7-48*f6*f0*f4*s1^6*s2^3+34* f6*f0*f3*s1^5*s2^4-2*f5^2*f1*s1^7*s2^2+3*f5*f1^2*s1^3*s2^6+30*f5*f1*f0*s1^2*s2^ 7-6*f5*f1*f4*s1^6*s2^3+22*f5*f0^2*s1*s2^8-18*f5*f0*f4*s1^5*s2^4+11*f5*f0*f3*s1^ 4*s2^5+5*f5*f4*f3*s1^8*s2-2*f1^2*f3*s1*s2^8+12*f1*f0*f4*s1*s2^8-2*f1*f0*f3*s2^9 +11*f0^2*f4*s2^9-10*f0*f4^2*s1^4*s2^5-5*f4^3*s1^8*s2+4*f2*f6*s1^6*s2-2*f2*f1*s1 *s2^6-2*f2*f0*s2^7+8*f6*f1*s1^5*s2^2+22*f6*f0*s1^4*s2^3+2*f5*f1*s1^4*s2^3+10*f5 *f0*s1^3*s2^4+6*f0*f4*s1^2*s2^5+2*f0*f3*s1*s2^6+2*f4^2*s1^6*s2; s8 := s1*s2^2+f6*s1^5-f1^2*f4*s1*s2^8-f5*f0^2*s2^9-f1*f3*s1*s2^6-5*f2^3*s1*s2^8 -f0*f3^2*s1*s2^8-f2*s1*s2^4+f5*s1^4*s2-10*f2^2*f6*s1^5*s2^4+11*f2*f6^2*s1^9+12* f2*f6*f5*s1^8*s2-18*f2*f6*f1*s1^4*s2^5-48*f2*f6*f0*s1^3*s2^6-12*f2*f6*f4*s1^7* s2^2-6*f2*f5*f1*s1^3*s2^6-24*f2*f5*f0*s1^2*s2^7+5*f2*f1*f3*s1*s2^8-16*f2*f0*f4* s1*s2^8+22*f6^2*f1*s1^8*s2+44*f6^2*f0*s1^7*s2^2+30*f6*f5*f1*s1^7*s2^2+76*f6*f5* f0*s1^6*s2^3-2*f6*f5*f3*s1^9+8*f6*f1*f0*s1^2*s2^7-8*f6*f1*f4*s1^6*s2^3+11*f6*f1 *f3*s1^5*s2^4-7*f6*f0^2*s1*s2^8+6*f6*f0*f4*s1^5*s2^4+34*f6*f0*f3*s1^4*s2^5+5*f6 *f4^2*s1^9-6*f6*f4*f3*s1^8*s2+3*f5^2*f1*s1^6*s2^3+17*f5^2*f0*s1^5*s2^4-2*f5^2* f3*s1^8*s2-2*f5*f1^2*s1^2*s2^7+2*f5*f1*f0*s1*s2^8-2*f5*f1*f4*s1^5*s2^4+8*f5*f0* f4*s1^4*s2^5+8*f5*f0*f3*s1^3*s2^6+5*f5*f4^2*s1^8*s2+2*f2^2*s1*s2^6+6*f2*f6*s1^5 *s2^2+10*f6*f1*s1^4*s2^3+22*f6*f0*s1^3*s2^4-2*f6*f4*s1^7+2*f6*f3*s1^6*s2+2*f5* f1*s1^3*s2^4+8*f5*f0*s1^2*s2^5-2*f5*f4*s1^6*s2+4*f0*f4*s1*s2^6; s9 := -5*f2^3*s2^9+s2^3-f2*s2^5+2*f2^2*s2^7+f6^2*f3*s1^9-f6*f0^2*s2^9-f2*f5*f4* s1^5*s2^4-f1^2*f4*s2^9-f6*f5*f3*s1^8*s2+f2*f5*s1^3*s2^4+f6*f5*s1^7+f5^2*s1^6*s2 -f0*f3^2*s2^9+f3*s1*s2^4-14*f2^2*f6*s1^4*s2^5-2*f2^2*f5*s1^3*s2^6+4*f2^2*f4*s1^ 2*s2^7+5*f2^2*f3*s1*s2^8+31*f2*f6^2*s1^8*s2+38*f2*f6*f5*s1^7*s2^2-24*f2*f6*f1* s1^3*s2^6-56*f2*f6*f0*s1^2*s2^7+10*f2*f6*f3*s1^5*s2^4-f1*f3*s2^7+4*f2*f5^2*s1^6 *s2^3-16*f2*f5*f1*s1^2*s2^7-35*f2*f5*f0*s1*s2^8-3*f2*f5*f3*s1^4*s2^5-6*f2*f1*f4 *s1*s2^8+5*f2*f1*f3*s2^9-22*f2*f0*f4*s2^9+2*f2*f4^2*s1^4*s2^5+64*f6^2*f1*s1^7* s2^2+148*f6^2*f0*s1^6*s2^3+90*f6*f5*f1*s1^6*s2^3+231*f6*f5*f0*s1^5*s2^4-3*f6*f5 *f4*s1^9+8*f6*f1^2*s1^2*s2^7+22*f6*f1*f0*s1*s2^8+24*f6*f1*f4*s1^5*s2^4+39*f6*f1 *f3*s1^4*s2^5+120*f6*f0*f4*s1^4*s2^5+96*f6*f0*f3*s1^3*s2^6+9*f6*f4^2*s1^8*s2-12 *f6*f4*f3*s1^7*s2^2+4*f6*f3^2*s1^6*s2^3+19*f5^2*f1*s1^5*s2^4+66*f5^2*f0*s1^4*s2 ^5-3*f5^2*f4*s1^8*s2-3*f5^2*f3*s1^7*s2^2+5*f5*f1*f0*s2^9+7*f5*f1*f4*s1^4*s2^5+6 *f5*f1*f3*s1^3*s2^6+70*f5*f0*f4*s1^3*s2^6+34*f5*f0*f3*s1^2*s2^7+6*f5*f4^2*s1^7* s2^2-8*f5*f4*f3*s1^6*s2^3-2*f1*f4*f3*s1^2*s2^7-2*f1*f3^2*s1*s2^8+20*f0*f4^2*s1^ 2*s2^7+12*f0*f4*f3*s1*s2^8+4*f4^3*s1^6*s2^3+10*f2*f6*s1^4*s2^3-2*f2*f4*s1^2*s2^ 5-2*f2*f3*s1*s2^6+18*f6*f1*s1^3*s2^4+30*f6*f0*s1^2*s2^5-4*f6*f4*s1^6*s2+6*f6*f3 *s1^5*s2^2+7*f5*f1*s1^2*s2^5+13*f5*f0*s1*s2^6-3*f5*f4*s1^5*s2^2+3*f5*f3*s1^4*s2 ^3+2*f1*f4*s1*s2^6+6*f0*f4*s2^7-2*f4^2*s1^4*s2^3+3*f6*s1^4*s2+3*f5*s1^3*s2^2+2* f4*s1^2*s2^3; s10 := f0^2*s2^8-14*f4^3*s1^10+5*f4^2*s1^8-2*f4*s1^6+s1^4-2*f2*f5^2*s1^10-10*f2 ^2*f0*s1^2*s2^8-40*f2*f6*f4*s1^10-36*f2*f6*f0*s1^6*s2^4-2*f2*f5*f1*s1^6*s2^4-8* f2*f5*f0*s1^5*s2^5-4*f2*f0^2*s2^10-12*f2*f0*f4*s1^4*s2^6-2*f6^2*f0*s1^10+6*f6* f5*f1*s1^10+24*f6*f5*f0*s1^9*s2+6*f6*f1^2*s1^6*s2^4-32*f6*f1*f0*s1^5*s2^5-64*f6 *f1*f4*s1^9*s2+12*f6*f1*f3*s1^8*s2^2-140*f6*f0^2*s1^4*s2^6-116*f6*f0*f4*s1^8*s2 ^2+48*f6*f0*f3*s1^7*s2^3-2*f6*f3^2*s1^10-4*f5^2*f1*s1^9*s2-4*f5^2*f0*s1^8*s2^2- 6*f5*f1*f0*s1^4*s2^6-10*f5*f1*f4*s1^8*s2^2-48*f5*f0^2*s1^3*s2^7-32*f5*f0*f4*s1^ 7*s2^3+12*f5*f0*f3*s1^6*s2^4+12*f5*f4*f3*s1^10+4*f1*f0*f3*s1^2*s2^8-22*f0^2*f4* s1^2*s2^8-20*f0*f4^2*s1^6*s2^4+8*f2*f6*s1^8+4*f2*f0*s1^2*s2^6+16*f6*f1*s1^7*s2+ 36*f6*f0*s1^6*s2^2+2*f5*f1*s1^6*s2^2+8*f5*f0*s1^5*s2^3-2*f5*f3*s1^8+6*f0*f4*s1^ 4*s2^4-2*f0*s1^2*s2^4; s11 := 2*s1^3*s2+f5*s1^6-f6^2*f1*s1^10+f5^2*f1*s1^8*s2^2-f1*f3^2*s1^4*s2^6+f6* f3*s1^8-f0^2*f3*s2^10-f2*f3*s1^4*s2^4-2*f2^2*f5*s1^6*s2^4+5*f2^2*f1*s1^2*s2^8- 10*f2^2*f0*s1*s2^9+2*f2^2*f3*s1^4*s2^6+14*f2*f6*f5*s1^10+12*f2*f6*f1*s1^6*s2^4- 44*f2*f6*f0*s1^5*s2^5-56*f2*f6*f4*s1^9*s2+14*f2*f6*f3*s1^8*s2^2-f1*f0*s2^8+f1* f4*s1^4*s2^4+f1*s1^2*s2^4+f3*s1^4*s2^2-2*f2*f5^2*s1^9*s2-8*f2*f5*f1*s1^5*s2^5- 24*f2*f5*f0*s1^4*s2^6-6*f2*f5*f4*s1^8*s2^2+4*f2*f1*f0*s2^10-2*f2*f1*f4*s1^4*s2^ 6-24*f2*f0*f4*s1^3*s2^7+2*f2*f4*f3*s1^6*s2^4-10*f6^2*f0*s1^9*s2+40*f6*f5*f1*s1^ 9*s2+110*f6*f5*f0*s1^8*s2^2+34*f6*f1^2*s1^5*s2^5+54*f6*f1*f0*s1^4*s2^6-80*f6*f1 *f4*s1^8*s2^2+52*f6*f1*f3*s1^7*s2^3-152*f6*f0^2*s1^3*s2^7-128*f6*f0*f4*s1^7*s2^ 3+162*f6*f0*f3*s1^6*s2^4-4*f6*f4*f3*s1^10-2*f6*f3^2*s1^9*s2+20*f5^2*f0*s1^7*s2^ 3-3*f5^2*f3*s1^10+5*f5*f1^2*s1^4*s2^6+20*f5*f1*f0*s1^3*s2^7-20*f5*f1*f4*s1^7*s2 ^3+6*f5*f1*f3*s1^6*s2^4-57*f5*f0^2*s1^2*s2^8-28*f5*f0*f4*s1^6*s2^4+46*f5*f0*f3* s1^5*s2^5+9*f5*f4^2*s1^10+10*f5*f4*f3*s1^9*s2-2*f5*f3^2*s1^8*s2^2-2*f1^2*f3*s1^ 2*s2^8+10*f1*f0*f4*s1^2*s2^8+4*f1*f0*f3*s1*s2^9-2*f1*f4^2*s1^6*s2^4-26*f0^2*f4* s1*s2^9-20*f0*f4^2*s1^5*s2^5+12*f0*f4*f3*s1^4*s2^6-10*f4^3*s1^9*s2+5*f4^2*f3*s1 ^8*s2^2+16*f2*f6*s1^7*s2+2*f2*f5*s1^6*s2^2-2*f2*f1*s1^2*s2^6+4*f2*f0*s1*s2^7+32 *f6*f1*s1^6*s2^2+76*f6*f0*s1^5*s2^3+8*f5*f1*s1^5*s2^3+23*f5*f0*s1^4*s2^4-3*f5* f4*s1^8-2*f5*f3*s1^7*s2+12*f0*f4*s1^3*s2^5+4*f4^2*s1^7*s2-2*f4*f3*s1^6*s2^2-2* f0*s1*s2^5-2*f4*s1^5*s2; s12 := -f6*s1^6+s1^2*s2^2-f2*s1^2*s2^4-f0*s2^6-5*f2^3*s1^2*s2^8-10*f2^2*f6*s1^6 *s2^4-9*f2^2*f0*s2^10-2*f2^2*f4*s1^4*s2^6-13*f2*f6^2*s1^10-20*f2*f6*f1*s1^5*s2^ 5-76*f2*f6*f0*s1^4*s2^6-22*f2*f6*f4*s1^8*s2^2-f2*f5^2*s1^8*s2^2-6*f2*f5*f1*s1^4 *s2^6+5*f2*f1*f3*s1^2*s2^8-22*f2*f0*f4*s1^2*s2^8-2*f2*f4^2*s1^6*s2^4-28*f6^2*f1 *s1^9*s2-71*f6^2*f0*s1^8*s2^2+3*f6*f5*f3*s1^10+f6*f1^2*s1^4*s2^6+11*f6*f1*f3*s1 ^6*s2^4-71*f6*f0^2*s1^2*s2^8-76*f6*f0*f4*s1^6*s2^4+48*f6*f0*f3*s1^5*s2^5-9*f6* f4^2*s1^10-f6*f3^2*s1^8*s2^2-2*f5^2*f1*s1^7*s2^3+f5^2*f0*s1^6*s2^4-2*f5*f1^2*s1 ^3*s2^7-6*f5*f1*f4*s1^6*s2^4-28*f5*f0^2*s1*s2^9-20*f5*f0*f4*s1^5*s2^5+11*f5*f0* f3*s1^4*s2^6+5*f5*f4*f3*s1^8*s2^2-f1^2*f4*s1^2*s2^8+3*f1*f0*f3*s2^10-13*f0^2*f4 *s2^10-10*f0*f4^2*s1^4*s2^6-f0*f3^2*s1^2*s2^8-5*f4^3*s1^8*s2^2+2*f2^2*s1^2*s2^6 +6*f2*f6*s1^6*s2^2+3*f2*f0*s2^8+f2*f4*s1^4*s2^4+12*f6*f1*s1^5*s2^3+37*f6*f0*s1^ 4*s2^4+3*f6*f4*s1^8+2*f5*f1*s1^4*s2^4+12*f5*f0*s1^3*s2^5-f5*f3*s1^6*s2^2-f1*f3* s1^2*s2^6+6*f0*f4*s1^2*s2^6+2*f4^2*s1^6*s2^2-f4*s1^4*s2^2+f2*f5*f3*s1^6*s2^4+f1 *f4*f3*s1^4*s2^6-32*f2*f5*f0*s1^3*s2^7-32*f6*f1*f4*s1^7*s2^3; s13 := 2*s1*s2^3+f1*s2^6-f6^2*f3*s1^10+f5*f1^2*s1^2*s2^8-f5*f0^2*s2^10-f5*f3^2* s1^6*s2^4-10*f2^3*s1*s2^9+f2*f5*s1^4*s2^4-f6*f5*s1^8+f0*f3*s2^8-f4*f3*s1^4*s2^4 +f5*s1^4*s2^2+f3*s1^2*s2^4-20*f2^2*f6*s1^5*s2^5-2*f2^2*f5*s1^4*s2^6+9*f2^2*f1* s2^10+5*f2^2*f3*s1^2*s2^8-26*f2*f6^2*s1^9*s2+10*f2*f6*f5*s1^8*s2^2-28*f2*f6*f1* s1^4*s2^6-128*f2*f6*f0*s1^3*s2^7-24*f2*f6*f4*s1^7*s2^3+12*f2*f6*f3*s1^6*s2^4-20 *f2*f5*f1*s1^3*s2^7-80*f2*f5*f0*s1^2*s2^8-2*f2*f5*f4*s1^6*s2^4-6*f2*f1*f4*s1^2* s2^8+10*f2*f1*f3*s1*s2^9-56*f2*f0*f4*s1*s2^9-4*f2*f0*f3*s2^10+2*f2*f4*f3*s1^4* s2^6-57*f6^2*f1*s1^8*s2^2-152*f6^2*f0*s1^7*s2^3+20*f6*f5*f1*s1^7*s2^3+54*f6*f5* f0*s1^6*s2^4+4*f6*f5*f4*s1^10+4*f6*f5*f3*s1^9*s2+20*f6*f1^2*s1^3*s2^7+110*f6*f1 *f0*s1^2*s2^8-24*f6*f1*f4*s1^6*s2^4+46*f6*f1*f3*s1^5*s2^5-10*f6*f0^2*s1*s2^9-44 *f6*f0*f4*s1^5*s2^5+162*f6*f0*f3*s1^4*s2^6-10*f6*f4^2*s1^9*s2+5*f5^2*f1*s1^6*s2 ^4+34*f5^2*f0*s1^5*s2^5-2*f5^2*f3*s1^8*s2^2+40*f5*f1*f0*s1*s2^9-8*f5*f1*f4*s1^5 *s2^5+6*f5*f1*f3*s1^4*s2^6+12*f5*f0*f4*s1^4*s2^6+52*f5*f0*f3*s1^3*s2^7+5*f5*f4^ 2*s1^8*s2^2-2*f1^2*f4*s1*s2^9-3*f1^2*f3*s2^10+14*f1*f0*f4*s2^10-2*f1*f4^2*s1^4* s2^6-2*f1*f3^2*s1^2*s2^8+14*f0*f4*f3*s1^2*s2^8-2*f0*f3^2*s1*s2^9+2*f4^2*f3*s1^6 *s2^4+4*f2^2*s1*s2^7+12*f2*f6*s1^5*s2^3-3*f2*f1*s2^8-2*f2*f3*s1^2*s2^6+23*f6*f1 *s1^4*s2^4+76*f6*f0*s1^3*s2^5+4*f6*f4*s1^7*s2+8*f5*f1*s1^3*s2^5+32*f5*f0*s1^2* s2^6-2*f5*f4*s1^6*s2^2+2*f1*f4*s1^2*s2^6-2*f1*f3*s1*s2^7+16*f0*f4*s1*s2^7-2*f2* s1*s2^5-2*f6*s1^5*s2; s14 := s2^4-2*f2*s2^6-14*f2^3*s2^10+f6^2*s1^8+5*f2^2*s2^8-4*f6^2*f4*s1^10-20*f2 ^2*f6*s1^4*s2^6-22*f2*f6^2*s1^8*s2^2-32*f2*f6*f1*s1^3*s2^7-116*f2*f6*f0*s1^2*s2 ^8-12*f2*f6*f4*s1^6*s2^4-10*f2*f5*f1*s1^2*s2^8-64*f2*f5*f0*s1*s2^9+12*f2*f1*f3* s2^10-40*f2*f0*f4*s2^10-48*f6^2*f1*s1^7*s2^3-140*f6^2*f0*s1^6*s2^4-6*f6*f5*f1* s1^6*s2^4-32*f6*f5*f0*s1^5*s2^5+4*f6*f5*f3*s1^8*s2^2-4*f6*f1^2*s1^2*s2^8+24*f6* f1*f0*s1*s2^9-8*f6*f1*f4*s1^5*s2^5+12*f6*f1*f3*s1^4*s2^6-2*f6*f0^2*s2^10-36*f6* f0*f4*s1^4*s2^6+48*f6*f0*f3*s1^3*s2^7-10*f6*f4^2*s1^8*s2^2+6*f5^2*f0*s1^4*s2^6- 4*f5*f1^2*s1*s2^9+6*f5*f1*f0*s2^10-2*f5*f1*f4*s1^4*s2^6+12*f5*f0*f3*s1^2*s2^8-2 *f1^2*f4*s2^10-2*f0*f3^2*s2^10+6*f2*f6*s1^4*s2^4+8*f6*f1*s1^3*s2^5+36*f6*f0*s1^ 2*s2^6+4*f6*f4*s1^6*s2^2+2*f5*f1*s1^2*s2^6+16*f5*f0*s1*s2^7-2*f1*f3*s2^8+8*f0* f4*s2^8-2*f6*s1^4*s2^2; s15 := 4*f6*s1^6+4*f2*s1^2*s2^4+f5^2*s1^8+f1^2*s2^8+4*f0*s2^6+f3^2*s1^4*s2^4+20 *f2^3*s1^2*s2^8+40*f2^2*f6*s1^6*s2^4-8*f2^2*f5*s1^5*s2^5+20*f2^2*f1*s1*s2^9+36* f2^2*f0*s2^10+8*f2^2*f4*s1^4*s2^6+8*f2^2*f3*s1^3*s2^7+52*f2*f6^2*s1^10+56*f2*f6 *f5*s1^9*s2+128*f2*f6*f1*s1^5*s2^5+272*f2*f6*f0*s1^4*s2^6+24*f2*f6*f4*s1^8*s2^2 +56*f2*f6*f3*s1^7*s2^3+8*f2*f5^2*s1^8*s2^2-16*f2*f5*f4*s1^7*s2^3-4*f2*f1^2*s2^ 10-16*f2*f1*f4*s1^3*s2^7-26*f2*f1*f3*s1^2*s2^8+24*f2*f0*f4*s1^2*s2^8-12*f2*f0* f3*s1*s2^9+8*f2*f4^2*s1^6*s2^4+4*f2*f4*f3*s1^5*s2^5-2*f2*f3^2*s1^4*s2^6+108*f6^ 2*f1*s1^9*s2+252*f6^2*f0*s1^8*s2^2+136*f6*f5*f1*s1^8*s2^2+352*f6*f5*f0*s1^7*s2^ 3-10*f6*f5*f3*s1^10+108*f6*f1^2*s1^4*s2^6+352*f6*f1*f0*s1^3*s2^7+118*f6*f1*f3* s1^6*s2^4+252*f6*f0^2*s1^2*s2^8+272*f6*f0*f4*s1^6*s2^4+276*f6*f0*f3*s1^5*s2^5+ 36*f6*f4^2*s1^10-12*f6*f4*f3*s1^9*s2+6*f6*f3^2*s1^8*s2^2+36*f5^2*f1*s1^7*s2^3+ 108*f5^2*f0*s1^6*s2^4-4*f5^2*f4*s1^10-8*f5^2*f3*s1^9*s2+36*f5*f1^2*s1^3*s2^7+ 136*f5*f1*f0*s1^2*s2^8+36*f5*f1*f3*s1^5*s2^5+108*f5*f0^2*s1*s2^9+128*f5*f0*f4* s1^5*s2^5+118*f5*f0*f3*s1^4*s2^6+20*f5*f4^2*s1^9*s2-26*f5*f4*f3*s1^8*s2^2-4*f5* f3^2*s1^7*s2^3+8*f1^2*f4*s1^2*s2^8-8*f1^2*f3*s1*s2^9+56*f1*f0*f4*s1*s2^9-10*f1* f0*f3*s2^10-8*f1*f4^2*s1^5*s2^5-4*f1*f3^2*s1^3*s2^7+52*f0^2*f4*s2^10+40*f0*f4^2 *s1^4*s2^6+56*f0*f4*f3*s1^3*s2^7+6*f0*f3^2*s1^2*s2^8+20*f4^3*s1^8*s2^2+8*f4^2* f3*s1^7*s2^3-2*f4*f3^2*s1^6*s2^4-8*f2^2*s1^2*s2^6+8*f2*f6*s1^6*s2^2+8*f2*f5*s1^ 5*s2^3-8*f2*f1*s1*s2^7-12*f2*f0*s2^8-4*f2*f4*s1^4*s2^4-4*f2*f3*s1^3*s2^5+16*f6* f1*s1^5*s2^3+12*f6*f0*s1^4*s2^4-12*f6*f4*s1^8+4*f6*f3*s1^7*s2+18*f5*f1*s1^4*s2^ 4+16*f5*f0*s1^3*s2^5-8*f5*f4*s1^7*s2+6*f5*f3*s1^6*s2^2+8*f1*f4*s1^3*s2^5+6*f1* f3*s1^2*s2^6+8*f0*f4*s1^2*s2^6+4*f0*f3*s1*s2^7-8*f4^2*s1^6*s2^2-4*f4*f3*s1^5*s2 ^3+4*f5*s1^5*s2+4*f1*s1*s2^5+4*f4*s1^4*s2^2+4*f3*s1^3*s2^3-2*f2*f5*f3*s1^6*s2^4 -2*f1*f4*f3*s1^4*s2^6+64*f2*f5*f0*s1^3*s2^7+64*f6*f1*f4*s1^7*s2^3; # Then (k1,k2,k3,k4) = (s5, f1*s14+f3*s12+2*s4+f5*s10, s3, s0). k1ps := subs(f6=ff6, f5=ff5, f4=ff4, f3=ff3, f2=ff2, f1=ff1, f0=ff0, s5); k2ps := expand(subs(f6=ff6, f5=ff5, f4=ff4, f3=ff3, f2=ff2, f1=ff1, f0=ff0, f1*s14+f3*s12+2*s4+f5*s10 )); k3ps := subs(f6=ff6, f5=ff5, f4=ff4, f3=ff3, f2=ff2, f1=ff1, f0=ff0, s3); k4ps := subs(f6=ff6, f5=ff5, f4=ff4, f3=ff3, f2=ff2, f1=ff1, f0=ff0, s0); # The following checks that, as we would expect, # the degree 4,6,8,10 terms vanish in kummeqn. kummeqnps := expand(subs(k1=k1ps, k2=k2ps, k3=k3ps, k4=k4ps, kummeqn)); coeff( subs(s1=z*s1, s2=z*s2, kummeqnps), z, 4 ); coeff( subs(s1=z*s1, s2=z*s2, kummeqnps), z, 6 ); coeff( subs(s1=z*s1, s2=z*s2, kummeqnps), z, 8 ); coeff( subs(s1=z*s1, s2=z*s2, kummeqnps), z, 10 ); coeff( subs(s1=z*s1, s2=z*s2, kummeqnps), z, 12 ); l1ps := expand(subs(k1=k1ps, k2=k2ps, k3=k3ps, k4=k4ps, l1)); l2ps := expand(subs(k1=k1ps, k2=k2ps, k3=k3ps, k4=k4ps, l2)); l3ps := expand(subs(k1=k1ps, k2=k2ps, k3=k3ps, k4=k4ps, l3)); l4ps := expand(subs(k1=k1ps, k2=k2ps, k3=k3ps, k4=k4ps, l4)); part1 := expand( (l2^2-4*l1*l3)*l4^2 - k4^16*kummeqn); part1ps := expand( subs(k1=k1ps, k2=k2ps, k3=k3ps, k4=k4ps, part1) ); # The following is the initial degree 6 portions of part1ps. part1psd6 := coeff( subs(s1=z*s1, s2=z*s2, part1ps), z, 6 ); # This gave: # part1psd6 := -13*s1^6-3072*s2^6-5440*s1^2*s2^4+6656*s1*s2^5 # -860*s1^4*s2^2+2720*s1^3*s2^3+136*s1^5*s2; # The following are the initial degree 2 portions of l1ps,l2ps,l3ps. l1psd2 := coeff( subs(s1=z*s1, s2=z*s2, l1ps), z, 2 ); l2psd2 := coeff( subs(s1=z*s1, s2=z*s2, l2ps), z, 2 ); l3psd2 := coeff( subs(s1=z*s1, s2=z*s2, l3ps), z, 2 ); # The above gave: s2^2, 2*s1*s2, s1^2, respectively. # Hence the following are all zero: l1psd2*l2psd2^2 - 4*l1psd2^2*l3psd2; l2psd2^3 - 4*l1psd2*l2psd2*l3psd2; l2psd2^2*l3psd2 - 4*l1psd2*l3psd2^2; # (the above is likely why we didn't resolve the quartic before). # The following give, respectively: # s2^6, s1*s2^5, s1^2*s2^4, s1^3*s2^3, s1^4*s2^2, s1^5*s2, s1^6. l1psd2^3; (1/2)*l1psd2^2*l2psd2; l1psd2^2*l3psd2; (1/2)*l1psd2*l2psd2*l3psd2; l1psd2*l3psd2^2; (1/2)*l2psd2*l3psd2^2; l3psd2^3; # We shall now create part2 which makes the degree 6 terms vanish. part2ps := part1ps - coeff(part1psd6,s2,6)*l1ps^3*l4ps - coeff(coeff(part1psd6,s2,5),s1,1)*(1/2)*l1ps^2*l2ps*l4ps - coeff(coeff(part1psd6,s2,4),s1,2)*l1ps^2*l3ps*l4ps - coeff(coeff(part1psd6,s2,3),s1,3)*(1/2)*l1ps*l2ps*l3ps*l4ps - coeff(coeff(part1psd6,s2,2),s1,4)*l1ps*l3ps^2*l4ps - coeff(coeff(part1psd6,s2,1),s1,5)*(1/2)*l2ps*l3ps^2*l4ps - coeff(part1psd6,s1,6)*l3ps^3*l4ps - t1*(l1ps*l2ps^2 - 4*l1ps^2*l3ps)*l4ps - t2*(l2ps^3 - 4*l1ps*l2ps*l3ps)*l4ps - t3*(l2ps^2*l3ps - 4*l1ps*l3ps^2)*l4ps; # This was: # part2ps := part1ps # - l4ps*( - 3072*l1ps^3 # + 3328*l1ps^2*l2ps # - 5440*l1ps^2*l3ps # + 1360*l1ps*l2ps*l3ps # - 860*l1ps*l3ps^2 # + 68*l2ps*l3ps^2 # - 13*l3ps^3 # + t1*(l1ps*l2ps^2 - 4*l1ps^2*l3ps) # + t2*(l2ps^3 - 4*l1ps*l2ps*l3ps) # + t3*(l2ps^2*l3ps - 4*l1ps*l3ps^2) ); # # part2 := part1 - l4*( - 3072*l1^3 + 3328*l1^2*l2 - 5440*l1^2*l3 + 1360*l1*l2*l3 - 860*l1*l3^2 + 68*l2*l3^2 - 13*l3^3 + t1*(l1*l2^2 - 4*l1^2*l3) + t2*(l2^3 - 4*l1*l2*l3) + t3*(l2^2*l3 - 4*l1*l3^2) ); part2 := expand(part2); # # # # # part2ps := expand(part2ps); # The following is zero: part2psd6 := coeff( subs(s1=z*s1, s2=z*s2, part2ps), z, 6 ); # Now focus on the degree 8 part: part2psd8 := coeff( subs(s1=z*s1, s2=z*s2, part2ps), z, 8 ); # # The following are all zero: l1psd2^2*l2psd2^2 - 4*l1psd2^3*l3psd2; l1psd2*l2psd2^3 - 4*l1psd2^2*l2psd2*l3psd2; l1psd2*l2psd2^2*l3psd2 - 4*l1psd2^2*l3psd2^2; l2psd2^4 - 4*l1psd2*l2psd2^2*l3psd2; l2psd2^3*l3psd2 - 4*l1psd2*l2psd2*l3psd2^2; l2psd2^2*l3psd2^2 - 4*l1psd2*l3psd2^3; # The following give, respectively: # s2^8, s1*s2^7, s1^2*s2^6, s1^3*s2^5, s1^4*s2^4, # s1^5*s2^3, s1^6*s2^2, s1^7*s2, s1^8. l1psd2^4; (1/2)*l1psd2^3*l2psd2; l1psd2^3*l3psd2; (1/2)*l1psd2^2*l2psd2*l3psd2; l1psd2^2*l3psd2^2; (1/2)*l1psd2*l2psd2*l3psd2^2; l1psd2*l3psd2^3; (1/2)*l2psd2*l3psd2^3; l3psd2^4; # part3ps := part2ps - coeff(part2psd8,s2,8)*l1ps^4 - coeff(coeff(part2psd8,s2,7),s1,1)*(1/2)*l1ps^3*l2ps - coeff(coeff(part2psd8,s2,6),s1,2)*l1ps^3*l3ps - coeff(coeff(part2psd8,s2,5),s1,3)*(1/2)*l1ps^2*l2ps*l3ps - coeff(coeff(part2psd8,s2,4),s1,4)*l1ps^2*l3ps^2 - coeff(coeff(part2psd8,s2,3),s1,5)*(1/2)*l1ps*l2ps*l3ps^2 - coeff(coeff(part2psd8,s2,2),s1,6)*l1ps*l3ps^3 - coeff(coeff(part2psd8,s2,1),s1,7)*(1/2)*l2ps*l3ps^3 - coeff(part2psd8,s1,8)*l3ps^4 - t4*(l1ps^2*l2ps^2 - 4*l1ps^3*l3ps) - t5*(l1ps*l2ps^3 - 4*l1ps^2*l2ps*l3ps) - t6*(l1ps*l2ps^2*l3ps - 4*l1ps^2*l3ps^2) - t7*(l2ps^4 - 4*l1ps*l2ps^2*l3ps) - t8*(l2ps^3*l3ps - 4*l1ps*l2ps*l3ps^2) - t9*(l2ps^2*l3ps^2 - 4*l1ps*l3ps^3); # This was: # part3ps := part2ps # - ( (3072*t1+5931008)*l1ps^4 # + (-7331840+3072*t2-3328*t1)*l1ps^3*l2ps # + (14735360-13312*t2+5440*t1+3072*t3)*l1ps^3*l3ps # + (-4715008+5440*t2-1360*t1-3328*t3)*l1ps^2*l2ps*l3ps # + (4051840-5440*t2+860*t1+5440*t3)*l1ps^2*l3ps^2 # + (-593152+860*t2-68*t1-1360*t3)*l1ps*l2ps*l3ps^2 # + (259440-272*t2+13*t1+860*t3)*l1ps*l3ps^3 # + (-17640+13*t2-68*t3)*l2ps*l3ps^3 # + (13*t3+3043)*l3ps^4 # + t4*(l1ps^2*l2ps^2 - 4*l1ps^3*l3ps) # + t5*(l1ps*l2ps^3 - 4*l1ps^2*l2ps*l3ps) # + t6*(l1ps*l2ps^2*l3ps - 4*l1ps^2*l3ps^2) # + t7*(l2ps^4 - 4*l1ps*l2ps^2*l3ps) # + t8*(l2ps^3*l3ps - 4*l1ps*l2ps*l3ps^2) # + t9*(l2ps^2*l3ps^2 - 4*l1ps*l3ps^3) ); part3 := part2 - ( (3072*t1+5931008)*l1^4 + (-7331840+3072*t2-3328*t1)*l1^3*l2 + (14735360-13312*t2+5440*t1+3072*t3)*l1^3*l3 + (-4715008+5440*t2-1360*t1-3328*t3)*l1^2*l2*l3 + (4051840-5440*t2+860*t1+5440*t3)*l1^2*l3^2 + (-593152+860*t2-68*t1-1360*t3)*l1*l2*l3^2 + (259440-272*t2+13*t1+860*t3)*l1*l3^3 + (-17640+13*t2-68*t3)*l2*l3^3 + (13*t3+3043)*l3^4 + t4*(l1^2*l2^2 - 4*l1^3*l3) + t5*(l1*l2^3 - 4*l1^2*l2*l3) + t6*(l1*l2^2*l3 - 4*l1^2*l3^2) + t7*(l2^4 - 4*l1*l2^2*l3) + t8*(l2^3*l3 - 4*l1*l2*l3^2) + t9*(l2^2*l3^2 - 4*l1*l3^3) ); part3 := expand(part3); # # # part3ps := expand(part3ps); # The following is zero, as expected. part3psd8 := coeff( subs(s1=z*s1, s2=z*s2, part3ps), z, 8 ); # # Now consider the degree 10 terms. part3psd10 := coeff( subs(s1=z*s1, s2=z*s2, part3ps), z, 10 ); eq0 := coeff( part3psd10, s2, 10); eq1 := coeff( coeff( part3psd10, s2, 9), s1, 1); eq2 := coeff( coeff( part3psd10, s2, 8), s1, 2); eq3 := coeff( coeff( part3psd10, s2, 7), s1, 3); eq4 := coeff( coeff( part3psd10, s2, 6), s1, 4); eq5 := coeff( coeff( part3psd10, s2, 5), s1, 5); eq6 := coeff( coeff( part3psd10, s2, 4), s1, 6); eq7 := coeff( coeff( part3psd10, s2, 3), s1, 7); eq8 := coeff( coeff( part3psd10, s2, 2), s1, 8); eq9 := coeff( coeff( part3psd10, s2, 1), s1, 9); eq10 := coeff( part3psd10, s1, 10); subt := solve( {eq0,eq1,eq2,eq3,eq4,eq5,eq6,eq7,eq8,eq9,eq10}, {t1,t2,t3,t4,t5,t6,t7,t8,t9} ); # This gave: # subt := {t1 = -2944, t2 = 224, t3 = -216, t4 = -1879040, # t5 = 225280, t6 = -234240-4*t7, t7 = t7, t8 = 2560, t9 = -1600}; expand( subs( op(subt), part3psd10 ) ); part3subt1 := expand( subs( op(subt), part3 ) ); factor( coeff( part3subt1, k4, 17) ); part3subt2 := expand( part3subt1 - simplify(%/(k2^2-4*k1*k3))*k4^15*kummeqn ); factor( coeff( part3subt2, k4, 16) ); part3subt3 := expand( part3subt2 - simplify(%/(k2^2-4*k1*k3))*k4^14*kummeqn ); factor( coeff( part3subt3, k4, 15) ); part3subt4 := expand( part3subt3 - simplify(%/(k2^2-4*k1*k3))*k4^13*kummeqn ); factor( coeff( part3subt4, k4, 14) ); # The following finally forces the value of t7: t7soln := solve(coeff(%,k3,6)); # This gave: t7soln := -2560; part3subt := subs( op(subt), part3 ); part3subt := subs( t7 = t7soln, part3subt ); factorsofpart3subt := [factors(part3subt)]; # The following is a constant, proving the part3subt has kummeqn as a factor. simplify(factorsofpart3subt[1][2][2][1]/kummeqn); # # Now, subst the following: # t1 = -2944, t2 = 224, t3 = -216, t4 = -1879040, t5 = 225280, # t6 = -224000, t7 = -2560, t8 = 2560, t9 = -1600, # into the following: quarticl := subs( t1 = -2944, t2 = 224, t3 = -216, t4 = -1879040, t5 = 225280, t6 = -224000, t7 = -2560, t8 = 2560, t9 = -1600, (l2^2-4*l1*l3)*l4^2 - l4*( - 3072*l1^3 + 3328*l1^2*l2 - 5440*l1^2*l3 + 1360*l1*l2*l3 - 860*l1*l3^2 + 68*l2*l3^2 - 13*l3^3 + t1*(l1*l2^2 - 4*l1^2*l3) + t2*(l2^3 - 4*l1*l2*l3) + t3*(l2^2*l3 - 4*l1*l3^2) ) - ( (3072*t1+5931008)*l1^4 + (-7331840+3072*t2-3328*t1)*l1^3*l2 + (14735360-13312*t2+5440*t1+3072*t3)*l1^3*l3 + (-4715008+5440*t2-1360*t1-3328*t3)*l1^2*l2*l3 + (4051840-5440*t2+860*t1+5440*t3)*l1^2*l3^2 + (-593152+860*t2-68*t1-1360*t3)*l1*l2*l3^2 + (259440-272*t2+13*t1+860*t3)*l1*l3^3 + (-17640+13*t2-68*t3)*l2*l3^3 + (13*t3+3043)*l3^4 + t4*(l1^2*l2^2 - 4*l1^3*l3) + t5*(l1*l2^3 - 4*l1^2*l2*l3) + t6*(l1*l2^2*l3 - 4*l1^2*l3^2) + t7*(l2^4 - 4*l1*l2^2*l3) + t8*(l2^3*l3 - 4*l1*l2*l3^2) + t9*(l2^2*l3^2 - 4*l1*l3^3) ) ); simplify([factors(quarticl)][1][2][2][1]/kummeqn); quarticll := subs( t1 = -2944, t2 = 224, t3 = -216, t4 = -1879040, t5 = 225280, t6 = -224000, t7 = -2560, t8 = 2560, t9 = -1600, (ll2^2-4*ll1*ll3)*ll4^2 - ll4*( - 3072*ll1^3 + 3328*ll1^2*ll2 - 5440*ll1^2*ll3 + 1360*ll1*ll2*ll3 - 860*ll1*ll3^2 + 68*ll2*ll3^2 - 13*ll3^3 + t1*(ll1*ll2^2 - 4*ll1^2*ll3) + t2*(ll2^3 - 4*ll1*ll2*ll3) + t3*(ll2^2*ll3 - 4*ll1*ll3^2) ) - ( (3072*t1+5931008)*ll1^4 + (-7331840+3072*t2-3328*t1)*ll1^3*ll2 + (14735360-13312*t2+5440*t1+3072*t3)*ll1^3*ll3 + (-4715008+5440*t2-1360*t1-3328*t3)*ll1^2*ll2*ll3 + (4051840-5440*t2+860*t1+5440*t3)*ll1^2*ll3^2 + (-593152+860*t2-68*t1-1360*t3)*ll1*ll2*ll3^2 + (259440-272*t2+13*t1+860*t3)*ll1*ll3^3 + (-17640+13*t2-68*t3)*ll2*ll3^3 + (13*t3+3043)*ll3^4 + t4*(ll1^2*ll2^2 - 4*ll1^3*ll3) + t5*(ll1*ll2^3 - 4*ll1^2*ll2*ll3) + t6*(ll1*ll2^2*ll3 - 4*ll1^2*ll3^2) + t7*(ll2^4 - 4*ll1*ll2^2*ll3) + t8*(ll2^3*ll3 - 4*ll1*ll2*ll3^2) + t9*(ll2^2*ll3^2 - 4*ll1*ll3^3) ) ); quarticll := ll4^2*(ll2^2-4*ll1*ll3) + ll4*(3072*ll1^3-3328*ll1^2*ll2-6336*ll1^2*ll3+2944*ll1*ll2^2 -464*ll1*ll2*ll3-4*ll1*ll3^2-224*ll2^3+216*ll2^2*ll3 -68*ll2*ll3^2+13*ll3^3) + (3112960*ll1^4-3153920*ll1^3*ll2-2590720*ll1^3*ll3 +1879040*ll1^2*ll2^2-325120*ll1^2*ll2*ll3-22400*ll1^2*ll3^2 -225280*ll1*ll2^3+213760*ll1*ll2^2*ll3-83200*ll1*ll2*ll3^2 +19120*ll1*ll3^3+2560*ll2^4-2560*ll2^3*ll3+1600*ll2^2*ll3^2 +40*ll2*ll3^3-235*ll3^4); quarticl := l4^2*(l2^2-4*l1*l3) + l4*(3072*l1^3-3328*l1^2*l2-6336*l1^2*l3+2944*l1*l2^2 -464*l1*l2*l3-4*l1*l3^2-224*l2^3+216*l2^2*l3 -68*l2*l3^2+13*l3^3) + (3112960*l1^4-3153920*l1^3*l2-2590720*l1^3*l3 +1879040*l1^2*l2^2-325120*l1^2*l2*l3-22400*l1^2*l3^2 -225280*l1*l2^3+213760*l1*l2^2*l3-83200*l1*l2*l3^2 +19120*l1*l3^3+2560*l2^4-2560*l2^3*l3+1600*l2^2*l3^2 +40*l2*l3^3-235*l3^4); # The following returns a constant, showing that kummeqn divides quarticl. chk := simplify([factors(quarticl)][1][2][1][1]/kummeqn); # The following is an alternative (returns true). # chk1 := divide(quarticl, kummeqn); # # We recall that the following is the general kummer eqn. # We wish to recognise the above quarticl as a kummer eqn, # after a possible linear change in l1,l2,l3,l4. # R,S,T are as follows: # kumR := k2**2-4*k1*k3: # kumS := -4*k1**3*f0-2*k1**2*k2*f1-4*k1**2*k3*f2-2*k1*k2*k3*f3 # -4*k1*k3**2*f4-2*k2*k3**2*f5-4*k3**3*f6: # kumT := -4*k1**4*f0*f2+k1**4*f1**2-4*k1**3*k2*f0*f3 # -2*k1**3*k3*f1*f3-4*k1**2*k2**2*f0*f4+4*k1**2*k2*k3*f0*f5 # -4*k1**2*k2*k3*f1*f4-4*k1**2*k3**2*f0*f6+2*k1**2*k3**2*f1*f5 # -4*k1**2*k3**2*f2*f4+k1**2*k3**2*f3**2-4*k1*k2**3*f0*f5 # +8*k1*k2**2*k3*f0*f6-4*k1*k2**2*k3*f1*f5+4*k1*k2*k3**2*f1*f6 # -4*k1*k2*k3**2*f2*f5-2*k1*k3**3*f3*f5-4*k2**4*f0*f6-4*k2**3*k3*f1*f6 # -4*k2**2*k3**2*f2*f6-4*k2*k3**3*f3*f6-4*k3**4*f4*f6+k3**4*f5**2: # kummeqn := kumR*k4**2+kumS*k4+kumT: # Note that in the following portion of quarticl: # + l4*(3072*l1^3-3328*l1^2*l2-6336*l1^2*l3+2944*l1*l2^2 # -464*l1*l2*l3-4*l1*l3^2-224*l2^3+216*l2^2*l3 # -68*l2*l3^2+13*l3^3) # there are terms: l1*l2^2*l4, l2^3*l4, l2^2*l3*l4, # which do not appear in kummeqn. So, subst: # l4 = varl4 + u1*l1 + u2*l2 + u3*l3 # varquarticll := subs( ll4 = varll4 + u1*ll1 + u2*ll2 + u3*ll3, quarticll ); eqn := coeff( expand(varquarticll), varll4, 1); eqn1 := coeff(coeff(eqn,ll1,1),ll2,2); eqn2 := coeff(eqn,ll2,3); eqn3 := coeff(coeff(eqn,ll2,2),ll3,1); u1soln := solve(eqn1); u2soln := solve(eqn2); u3soln := solve(eqn3); newquarticll := subs( ll4 = varll4 + u1soln*ll1 + u2soln*ll2 + u3soln*ll3, quarticll ); fff0 := coeff(coeff(newquarticll,varll4,1),ll1,3)/(-4); fff1 := coeff(coeff(coeff(newquarticll,varll4,1),ll1,2),ll2,1)/(-2); fff2 := coeff(coeff(coeff(newquarticll,varll4,1),ll1,2),ll3,1)/(-4); fff3 := coeff(coeff(coeff(coeff(newquarticll,varll4,1),ll1,1),ll2,1),ll3,1)/(-2); fff4 := coeff(coeff(coeff(newquarticll,varll4,1),ll1,1),ll3,2)/(-4); fff5 := coeff(coeff(coeff(newquarticll,varll4,1),ll2,1),ll3,2)/(-2); fff6 := coeff(coeff(newquarticll,varll4,1),ll3,3)/(-4); chkum := subs( f0=fff0, f1=fff1, f2=fff2, f3=fff3, f4=fff4, f5=fff5, f6=fff6, k1=ll1, k2=ll2, k3=ll3, k4=varll4, kumR*k4**2+kumS*k4+kumT ); # The following is zero, proving that we now have exactly the # form of a Kummer eqn for y^2 = fff6*x^6 + ... + fff0. # # The following confirms that after replacing ll4 with varll4, # the Kummer equation precisely agrees with he new version of the quartic. chk := expand(chkum - newquarticll); newsextic := fff6*x^6 + fff5*x^5 + fff4*x^4 + fff3*x^3 + fff2*x^2 + fff1*x + fff0; newsextic := -13/4*x^6+34*x^5-215*x^4+680*x^3-1360*x^2+1664*x-768; DD := newsextic; gg6 := coeff(DD,x,6); gg5 := coeff(DD,x,5); gg4 := coeff(DD,x,4); gg3 := coeff(DD,x,3); gg2 := coeff(DD,x,2); gg1 := coeff(DD,x,1); gg0 := coeff(DD,x,0); kummeqnD := subs(f6=gg6, f5=gg5, f4=gg4, f3=gg3, f2=gg2, f1=gg1, f0=gg0, kummeqngeneral); l1 := l1; l2 := l2; l3 := l3; # The following adjusts l4, replacing it with varl4, as above. varl4 := expand(l4 - u1soln*l1 - u2soln*l2 - u3soln*l3); l4 := varl4; # The following confirms that the map (k1,k2,k3,k4) |--> (l1,l2,l3,l4) # is indeed from the Kummer of C to the Kummer of D. kummeqnDkk := subs(k1=kk1,k2=kk2,k3=kk3,k4=kk4, kummeqnD); chk2 := divide(expand(subs(kk1=l1,kk2=l2,kk3=l3,kk4=l4,kummeqnDkk)), expand(kummeqn)); # Our model for C is already quite nice: Cnice := C; # Now get the correct twist and check via 2-descent rank parities. # So far, we know that there is a map from the Kummer surface of: # C : y^2 = 256-640*x+560*x^2-200*x^3+25*x^4+x^5 # to the Kummer surface of: # D : y^2 = -13/4*x^6+34*x^5-215*x^4+680*x^3-1360*x^2+1664*x-768 # Note that C has the point (1,sqrt(2)), so that the quadratic twist: # C^(2) : y^2 = 2*(256-640*x+560*x^2-200*x^3+25*x^4+x^5) # has the point (1,2). # Also define: # D^(2) : y^2 = 2*(-13/4*x^6+34*x^5-215*x^4+680*x^3-1360*x^2+1664*x-768). # So, now take { (1,2), infty } on the Jacobian of C^(2), # map it to: P = { (1,sqrt(2)), infty } on the Jacobian of C. # Pk1 := 0; Pk2 := 1; Pk3 := 1; Pk4 := solve( subs( k1=Pk1, k2 = Pk2, k3 = Pk3, kummeqn ) )[1]; chk := simplify( subs( f6=ff6, f5=ff5, f4=ff4, f3=ff3, f2=ff2, f1=ff1, f0=ff0, k1=Pk1, k2=Pk2, k3=Pk3, k4=Pk4, kummeqn ) ); # The above gave: (0,1,1,1) on the Kummer surface of C. # Now put this into the quintic map from the Kummer surface of C # to the Kummer surface of D. Pl1 := subs( k1 = Pk1, k2 = Pk2, k3 = Pk3, k4 =Pk4, l1 ); Pl2 := subs( k1 = Pk1, k2 = Pk2, k3 = Pk3, k4 =Pk4, l2 ); Pl3 := subs( k1 = Pk1, k2 = Pk2, k3 = Pk3, k4 =Pk4, l3 ); Pl4 := subs( k1 = Pk1, k2 = Pk2, k3 = Pk3, k4 =Pk4, l4 ); # The above gave: ( 45/4, 36, 45, 9397 ) on the Kummer surface of D. # solve( x^2 - (Pl2/Pl1)*x + (Pl3/Pl1), x ); # The above give: x1 = 8/5+6/5*I, x2 = 8/5-6/5*I as the x-coordinates. # The final map from D to D^(2) does not affect the x-coordinates. # The correct twist should mean that something Q-rational in the # Jacobian of C^(2) should map to something Q-rational in the # Jacobian of D^(2). sqrt(subs( x = 8/5+6/5*I, 2*DD )); # This gives: 3672/125+304/125*I. # So that map from the Kummer of C^(2) to the Kummer of D^(2) # lifts to the Jacobian, since: # { (1,0), infty } |--> { (8/5+6/5*I,3672/125+304/125*I), conjugate }. # This means that we already have the correct twist, # and the Jacobian of C is (5,5)-isogenous over Q to the Jacobian of DD. # Note that the above curve D has a Weierstrass point! # Try to rewrite so that it obviously has two indep pts order 5, # and see if I can recognise a family. Also map it back. # First convert to quintic form, by mapping the Weierstrass pt to infinity. DD := -13/4*x^6+34*x^5-215*x^4+680*x^3-1360*x^2+1664*x-768; factor(%); # -1/4*(x-2)*(13*x^5-110*x^4+640*x^3-1440*x^2+2560*x-1536) subs(x=x+2, %); # -1/4*x*(13*(x+2)^5-110*(x+2)^4+640*(x+2)^3-1440*(x+2)^2+2560*x+3584) factor(%); # -1/4*x*(13*x^5+20*x^4+280*x^3+800*x^2+2000*x+1600) subs(x=1/x,%); # -1/4/x*(13/x^5+20/x^4+280/x^3+800/x^2+2000/x+1600) %*x^6; # -1/4*x^5*(13/x^5+20/x^4+280/x^3+800/x^2+2000/x+1600) factor(%); # -13/4-5*x-70*x^2-200*x^3-500*x^4-400*x^5 subs(x=-x,%); # -13/4+5*x-70*x^2+200*x^3-500*x^4+400*x^5 rr := subs(x=x/10,%)*25*100; # So, we'll take this as our nicer form of D. Dnice := rr; # rr := -8125+1250*x-1750*x^2+500*x^3-125*x^4+10*x^5 # We want to write rr in the form: # constant*(quadratic)^2 + 10*(x-s)^5 # ss := rr - 10*(x-s)^5; # ss := -8125+1250*x-1750*x^2+500*x^3-125*x^4+10*x^5-10*(x-s)^5; ss := expand(ss); # ss := -8125+1250*x-1750*x^2+500*x^3-125*x^4+50*x^4*s-100*x^3*s^2 # +100*x^2*s^3-50*x*s^4+10*s^5; # We want s to be chosen so that this is constant*(quadratic in x)^2, # so find the discrim in x, and solve for s, which will at least # force the quartic in x to have repeated roots. factor(discrim(ss,x)); # 31250000*(241056640625+4*s^14-101590000*s^7+18885375*s^8-3125500*s^9 # +429550*s^10-46600*s^11+3665*s^12-180*s^13-2110587500*s^5+487215000*s^6 # +74418593750*s^2-27931125000*s^3+8042609375*s^4-184459062500*s)*(s^2-5)^2 # We see that s := sqrt5 is worth a try. sqrt5 := RootOf(t^2 - 5); # sqrt5 := RootOf(_Z^2-5) factor( rr - 10*(x - sqrt5)^5 ); # 25*(-5+2*RootOf(_Z^2-5))*(x^2+15+4*RootOf(_Z^2-5))^2 # So, we now see that our curve D can be written as: # Dnice : y^2 = 25*(-5+2*sqrt5)*(x^2 + 15+4*sqrt5)^2 + 10*(x - sqrt5)^5 # Of course, since the curve is defined over Q, it can also be written: # Dnice : y^2 = 25*(-5-2*sqrt5)*(x^2 + 15-4*sqrt5)^2 + 10*(x + sqrt5)^5 # The points of order 5 in the kernel of the dual isogeny have support: # (sqrt5, 5*(5+15+4*sqrt5)*sqrt(-5+2*sqrt5), etc, # and are defined over the quartic field Q( sqrt(-5+2*sqrt5) ). # But note that this is just Q(zeta5), where zeta5 is a primitive 5th # root of unity. zeta5 := RootOf(t^4 + t^3 + t^2 + t + 1); # The following is zero. simplify( (zeta5 + zeta5^4)^2 + (zeta5 + zeta5^4) - 1 ); # solve(x^2 + x - 1); # 1/2*5^(1/2)-1/2, -1/2-1/2*5^(1/2) # This following is 5, explicitly giving Q(sqrt5) as a quadratic subfield # of Q(zeta5). (2*zeta5 + 2*zeta5^4 + 1)^2; # factor( x^2 - (-5 + 2*(2*zeta5 + 2*zeta5^4 + 1)) ); # -(x+1+2*RootOf(_Z^4+_Z^3+_Z^2+_Z+1)+2*RootOf(_Z^4+_Z^3+_Z^2+_Z+1)^3) # *(-x+1+2*RootOf(_Z^4+_Z^3+_Z^2+_Z+1)+2*RootOf(_Z^4+_Z^3+_Z^2+_Z+1)^3) # The above shows that there is a sqrt of -5+2*sqrt5 in Q(zeta5), # and clearly there is a C5xC5 of point of order 5, which are # each individually definied over Q(zeta5), and for which the # entire C5xC5 is defined over Q. Surely this is the kernel # of the dual isogeny (easy enough to check, by repeating this # file on Dnice, and getting back Cnice). factor(y^2 - subs( x = 2*zeta5 + 2*zeta5^4 + 1, Dnice )); # -(y+80+160*RootOf(_Z^4+_Z^3+_Z^2+_Z+1)+40*RootOf(_Z^4+_Z^3+_Z^2+_Z+1)^2 # +120*RootOf(_Z^4+_Z^3+_Z^2+_Z+1)^3)*(-y+80+160*RootOf(_Z^4+_Z^3+_Z^2+_Z+1) # +40*RootOf(_Z^4+_Z^3+_Z^2+_Z+1)^2+120*RootOf(_Z^4+_Z^3+_Z^2+_Z+1)^3) factor(y^2 - subs( x = -2*zeta5 - 2*zeta5^4 - 1, Dnice )); # -(-y-40-80*RootOf(_Z^4+_Z^3+_Z^2+_Z+1)-120*RootOf(_Z^4+_Z^3+_Z^2+_Z+1)^2 # +40*RootOf(_Z^4+_Z^3+_Z^2+_Z+1)^3)*(y-40-80*RootOf(_Z^4+_Z^3+_Z^2+_Z+1) # -120*RootOf(_Z^4+_Z^3+_Z^2+_Z+1)^2+40*RootOf(_Z^4+_Z^3+_Z^2+_Z+1)^3) # So, we see there are the points of order 5 on the Jacobian: # DT1 := {(1+2*zeta5+2*zeta5^4, 40*(2+4*zeta5+zeta5^2+3*zeta5^3), infty} # DT2 := {(-1-2*zeta5-2*zeta5^4, 40*(1+2*zeta5+3*zeta5^2-zeta5^3), infty} # The subgroup generated by { (x1,y1), infty } of order 5 is: # id, {(x1,y1),infty}, {(x1,y1),(x1,y1)}, {(x1,-y1),(x1,-y1)},{(x1,-y1),infty}. # # Now, consider the Cassels maps. # Our nice models Cnice and Dnice are as follows. # Cnice : y^2 = 5*(16 - 12*x + 3*x^2)^2 + (x - 4)^5 # = (16 - 20*x + 5*x^2)^2 + x^5 # Dnice : y^2 = 25*(-5+2*sqrt5)*(x^2 + 15+4*sqrt5)^2 + 10*(x - sqrt5)^5 # = 25*(-5-2*sqrt5)*(x^2 + 15-4*sqrt5)^2 + 10*(x + sqrt5)^5 # Let phi be the (5,5)-isogeny from Jac(Cnice) to Jac(Dnice). # Let phitilde be the dual (5,5)-isogeny from Jac(Dnice) to Jac(Cnice). # The Cassels map for Jac(Cnice)(Q)/ phitilde( Jac(Dnice)(Q) ) is: # Jac(Cnice)(Q)/ phitilde( Jac(Dnice)(Q) ) # --> Q(sqrt5)*/( Q(sqrt5)* )^5 x Q*/( Q* )^5 # : {(x1,y1),(x2,y2)} |--> # [ (y1 - sqrt(5)*(16 - 12*x1 + 3*x1^2))*(y2 - sqrt(5)*(16 - 12*x2 + 3*x2^2)), # (y1 - (16 - 20*x1 + 5*x1^2))*(y2 - (16 - 20*x2 + 5*x2^2)) ]. # The Cassels map for Jac(Dnice)(Q)/ phi( Jac(Cnice)(Q) ) is: # Jac(Dnice)(Q)/ phi( Jac(Cnice)(Q) ) # --> Q(sqrt(-5+2*sqrt5))*/( Q(sqrt(-5+2*sqrt5))* )^5 # x Q(sqrt(-5-2*sqrt5))*/( Q(sqrt(-5-2*sqrt5))* )^5 # : {(x1,y1),(x2,y2)} |--> # [ (y1 - 5*sqrt(-5+2*sqrt5)*(x1^2 + 15+4*sqrt5)) # *(y2 - 5*sqrt(-5+2*sqrt5)*(x2^2 + 15+4*sqrt5)), # (y1 - 5*sqrt(-5-2*sqrt5)*(x1^2 + 15-4*sqrt5)) # *(y2 - 5*sqrt(-5-2*sqrt5)*(x2^2 + 15-4*sqrt5)) ]. # (perhaps with a multiplicative translation, since non-monic). # As we have already observed, each of the quartic fields # Q(sqrt(-5+2*sqrt5)), Q(sqrt(-5-2*sqrt5)) can be taken to be Q(zeta5). # Note: just to be on the safe side, better to use Cmonic # before using the Cassels map, just to avoid confusion about # traslations, and to exploit that any element in the image # is then in the kernel of the norm map (mod 5th powers), # except that actually it is the Norm_{Q(zeta5)/Q(sqrt5)} which # will be a 5th power in Q(sqrt5), when the quintic becomes monic. # # Now consider what happens after a quadratic twist by r. Cnicer := r*Cnice; Cnicer := r*(256-640*x+560*x^2-200*x^3+25*x^4+x^5); Dnicer := r*Dnice; Dnicer := r*(-8125+1250*x-1750*x^2+500*x^3-125*x^4+10*x^5); # Cnicer : y^2 = r*( 5*(16 - 12*x + 3*x^2)^2 + (x - 4)^5 ) # = r*( (16 - 20*x + 5*x^2)^2 + x^5 ) # Dnicer : y^2 = r*( 25*(-5+2*sqrt5)*(x^2 + 15+4*sqrt5)^2 + 10*(x - sqrt5)^5 ) # = r*( 25*(-5-2*sqrt5)*(x^2 + 15-4*sqrt5)^2 + 10*(x + sqrt5)^5 ) # Make everything monic: Cmon := expand(subs( x = x/r, Cnicer)*r^4); Cmon := 256*r^5-640*x*r^4+560*x^2*r^3-200*x^3*r^2+25*x^4*r+x^5; Dmon := expand(subs( x = x/(10*r), Dnicer)*(10*r)^4); Dmon := -81250000*r^5+1250000*x*r^4-175000*x^2*r^3+5000*x^3*r^2-125*x^4*r+x^5; # Cmon: y^2 = r*5*(16*r^2 - 12*r*x + 3*x^2)^2 + (x - 4*r)^5 # = r*(16*r^2 - 20*r*x + 5*x^2)^2 + x^5 # Dmon: y^2= 25*r*(-5+2*sqrt5)*(x^2+(15+4*sqrt5)*100*r^2)^2 + (x-10*sqrt5*r)^5 # = 25*r*(-5-2*sqrt5)*(x^2+(15-4*sqrt5)*100*r^2)^2 + (x+10*sqrt5*r)^5 # # Note that the quadratic twist runs through the isogeny, # so we still have (5,5)-isogenies defined over Q # between Jac(Cmon) and Jac(Dmon). # # Let's see what difference is made by the quadratic twist. # The generators of the (5,5) on the Jacobian of Cmon are now: # T1 = { (4*r, 16*r^2*sqrt(5*r)), infty }. # T2 = { (0, 16*r^2*sqrt(r)), infty }. # As before the kernel of the isogeny (the subgroup generated # by T1,T2) is defined over Q. The entire list of 25 elements # in would require the quartic field Q( sqrt(5*r), sqrt(r) ) # but elts of are all defined over the quadratic field Q(sqrt(5*r)) # and elts of are all defined over the quadratic field Q(sqrt(r)). # The Cassels map for Jac(Cmon)(Q)/ phitilde( Jac(Dmon)(Q) ) is: # Jac(Cmon)(Q)/ phitilde( Jac(Dmon)(Q) ) # --> Q(sqrt(5*r))*/( Q(sqrt(5*r))* )^5 x Q(sqrt(r))*/( Q(sqrt(r))* )^5 # : {(x1,y1),(x2,y2)} |--> #[(y1-sqrt(5*r)*(16*r^2-12*x1*r+3*x1^2))*(y2-sqrt(5*r)*(16*r^2-12*x2*r+3*x2^2)), # (y1-sqrt(r)*(16*r^2-20*x1*r+5*x1^2))*(y2-sqrt(r)*(16*r^2-20*x2*r+5*x2^2))]. # So, this portion of the Selmer group computation would only # require working over quadratic number fields. # Note that anything in the image must have norm a 5th power in Q. This uses: # (y-sqrt(5*r)*(16*r^2-12*x*r+3*x^2))*(y+sqrt(5*r)*(16*r^2-12*x*r+3*x^2)) # = (x - 4*r)^5, # (y-sqrt(r)*(16*r^2-20*x*r+5*x^2))*(y+sqrt(r)*(16*r^2-20*x*r+5*x^2)) # = x^5. # # For Dmon, note that the generators of the (5,5) are now: # DT1 := { (10*sqrt5*r, 2000*r^2*sqrt(10*r*(-5+sqrt5))), infty } # DT2 := { (-10*sqrt5*r, 2000*r^2*sqrt(10*r*(-5-sqrt5))), infty } # As before the kernel of the isogeny (the subgroup generated # by DT1,DT2) is defined over Q. The entire list of 25 elements # in would require the degree 8 field Q( zeta5, sqrt(r) ) # but only requires the quartic field Q(sqrt(r*(-5+2*sqrt5))). # and only requires the quartic field Q(sqrt(r*(-5-2*sqrt5))). # # The Cassels map for Jac(Dmon)(Q)/ phi( Jac(Cmon)(Q) ) is: # Jac(Dmon)(Q)/ phi( Jac(Cmon)(Q) ) # --> Q(sqrt(r*(-5+2*sqrt5)))*/( Q(sqrt(r*(-5+2*sqrt5)))* )^5 # x Q(sqrt(r*(-5-2*sqrt5)))*/( Q(sqrt(r*(-5-2*sqrt5)))* )^5 # : {(x1,y1),(x2,y2)} |--> # [ (y1 - 5*sqrt(r*(-5+2*sqrt5))*(x1^2 + (15+4*sqrt5)*100*r^2)) # *(y2 - 5*sqrt(r*(-5+2*sqrt5))*(x2^2 + (15+4*sqrt5)*100*r^2)), # (y1 - 5*sqrt(r*(-5-2*sqrt5))*(x1^2 + (15-4*sqrt5)*100*r^2)) # *(y2 - 5*sqrt(r*(-5-2*sqrt5))*(x2^2 + (15-4*sqrt5)*100*r^2)) ]. # So, this portion of the Selmer group computation would only # require working over quartic number fields. # Since these are monic, any element in the image # is in the kernel of the norm map (mod 5th powers), # by which we mean that Norm_{Q(sqrt(r*(-5+2*sqrt5)))/Q(sqrt5)} # is a 5th power in Q(sqrt5). This uses: # (y - 5*sqrt(r*(-5+2*sqrt5))*(x^2 + (15+4*sqrt5)*100*r^2)) # *(y + 5*sqrt(r*(-5+2*sqrt5))*(x^2 + (15+4*sqrt5)*100*r^2)) # = (x-10*sqrt5*r)^5, # (y - 5*sqrt(r*(-5-2*sqrt5))*(x^2 + (15-4*sqrt5)*100*r^2)) # *(y + 5*sqrt(r*(-5-2*sqrt5))*(x^2 + (15-4*sqrt5)*100*r^2)) # = (x-10*sqrt5*r)^5, # # In summary, we should only even need to deal with at # worst quartic fields for the entire (5,5)-Selmer group computation, # even though the complete list of the elts of the kernel # of the dual isogeny generate a degree 8 number field. # # Note also that, if we merely want to obtain examples of Sha[5] # over Q, then it might be possible to do this using only # quadratic number fields. If there is no 5-torsion and the rank is 0 # (via a 2-descent in Magma) then Jac(Cmon)(Q)/5*Jac(Cmon)(Q) should # be the trivial group, and so J(Cmon)(Q)/phitilde(Jac(Dmon(Q))) # should also be the trivial group (being smaller than # Jac(Cmon)(Q)/5*Jac(Cmon)(Q), since phitilde(Jac(Dmon(Q))) # contains 5*Jac(Cmon)(Q)). So, there would be an element # in the 5-part of Sha over Q, if there is a nontrivial element in # Q(sqrt(5*r))*/( Q(sqrt(5*r))* )^5 x Q(sqrt(r))*/( Q(sqrt(r))* )^5 # which is mapped to locally (under the Cassels map) for all p. # Perhaps starts with r=q or r=-q, for q prime, and get conditions # on q such that there is an element mapped to everywhere locally # (and so that there is no 5-torsion) and then find an example # where Magma gives a 2-Selmer bound of 0. # Since we are only dealing with quadratic fields, maybe # a congruence class of primes is possible (see Flynn-Redmond). # # The following are monic versions, which might be better for # inputting into Magma. As an extra check I should, for lots of # values of r, check Magma's 2-Selmer bound on the rank of # the Jacobians of Cmon and Dmon, which should always have the # same parity (these are quintics, so Magma will be preforming a # true, not fake, complete 2-descent). # N.B. The file ed/stollsimple shows this is abs simple, using Stoll. Cmon := expand(subs( x = x/r, Cnicer)*r^4); Cmon := 256*r^5-640*x*r^4+560*x^2*r^3-200*x^3*r^2+25*x^4*r+x^5; Dmon := expand(subs( x = x/(10*r), Dnicer)*(10*r)^4); Dmon := -81250000*r^5+1250000*x*r^4-175000*x^2*r^3+5000*x^3*r^2-125*x^4*r+x^5; # # _ := PolynomialRing(Rationals()); # resultlist := []; # r := 1; # while r ne 8 do # Cmon := 256*r^5-640*x*r^4+560*x^2*r^3-200*x^3*r^2+25*x^4*r+x^5; # Dmon := -81250000*r^5+1250000*x*r^4-175000*x^2*r^3+5000*x^3*r^2-125*x^4*r+x^5; # rkCmon := TwoSelmerGroupData( Jacobian( HyperellipticCurve( Cmon ) ) ); # rkDmon := TwoSelmerGroupData( Jacobian( HyperellipticCurve( Dmon ) ) ); # Append( resultlist, [ r, rkCmon, rkDmon ] ); # r := -r; # Cmon := 256*r^5-640*x*r^4+560*x^2*r^3-200*x^3*r^2+25*x^4*r+x^5; # Dmon := -81250000*r^5+1250000*x*r^4-175000*x^2*r^3+5000*x^3*r^2-125*x^4*r+x^5; # rkCmon := TwoSelmerGroupData( Jacobian( HyperellipticCurve( Cmon ) ) ); # rkDmon := TwoSelmerGroupData( Jacobian( HyperellipticCurve( Dmon ) ) ); # Append( resultlist, [ r, rkCmon, rkDmon ] ); # r := -r; # r := r + 1; # end while; # # The above gave the following results, which are reassuring, # as the 2-Selmer rank bounds have the same parity (indeed are the # same) for Cmon and Dmon, consistent with the Jacobians # being isogenous over Q. # [ 1, 0, 0 ] # [ -1, 0, 0 ] # [ 2, 1, 1 ] # [ -2, 1, 1 ] # [ 3, 1, 1 ] # [ -3, 1, 1 ] # [ 4, 0, 0 ] # [ -4, 0, 0 ] # [ 5, 0, 0 ] # [ -5, 0, 0 ] # [ 6, 2, 2 ] # [ -6, 0, 0 ] # [ 7, 1, 1 ] # [ -7, 1, 1 ] # # Note: the case r=1 might be a good choice for performing # the entire descent to get a bound. Maybe this case even # has 5-part of Sha? # Maybe also I can exploit the point of order 5 in Jac(Cmon)(Q) # for the case r=1. Find its image under the Cassels map. # Choose r=p such that we maintain this being in the Selmer group, # but such that the 2-Selmer bound on the rank is still 0 # and such that there Jac(Cmon)(Q)[5] is the trivial group. # Then there should be 5-part of Sha. # Note how it should be possible to keep the 2-Selmer bound on # the rank still zero, with r equal to a cong class of prime, # since the theta (root of the quintic) will remain the same for all twists. # Note how this must induce another element in the Selmer group; # does this give some kind of reciprocity result? # # Here are a few preliminaries for the r=1 case. # First find the M's which contain the images of the the Cassels maps # (and recall also that the norm of anything in the image is a 5th power, # which cuts down further). All of these are in K*/(K*)^5, # so the number of generators is the 5-rank of each group. # The bad primes are: 2,5,infty. # For Q, <-1,2,5> is just: <2,5> (since -1 is already a 5th power). # For Q(sqrt5), with ring of integers Z[ (1+sqrt5)/2 ], classno=1, # unit group generated by -1 (torsion) and (1+sqrt5)/2 (fund unit), # and can ignore -1 since it is a 5th power. # 2 is inert (by DT, since x^2-x-1, minpoly of (1+sqrt5)/2, is irred mod 2). # 5 factors as sqrt5*sqrt5. So, can use: < (1+sqrt5)/2, 2, sqrt5 >. # Note: these have norms -1,2^2,-5, so it looks to me that we # can exclude everything except < (1+sqrt5)/2 >, using the # fact that norms must be 5th powers in Q. # For Q(zeta5). Note: for this direction of the isogeny, express # everything, the forms of the curve, Cassels maps, etc, all # in terms of zeta5; use: sqrt5 = 2*zeta5 + 2*zeta5^(-1) + 1, # so that zeta5 is a root of: x^2 + ((1-sqrt5)/2)*x + 1. # Note that the conjugates of zeta5 over Q(sqrt5)) are: zeta5, zeta5^(-1). # The ring of integers is Z[zeta5], classno=1, # the unit group is generated by -zeta5 (torsion) # and 1+zeta5 (fund unit), and might as well use zeta5 instead # of -zeta5, since -1 is a 5th power. Also, 2 is inert # by DT (since x^4+x^3+x^2+x+1 is irred mod 2), and # 5 factors as 5 = (1-zeta5)*(1-zeta5^2)*(1-zeta5^3)*(1-zeta5)^4, # and these are all associates (since (1-zeta5^i)/(1-zeta) is # just 1 + zeta5 + ... + zeta5^{i-1}, which has norm 1 and is a unit). # So, we can use: . These have norms over Q: # 1,1,2^4,5. They have norms Q(zeta5)/Q(sqrt5) given by: # Norm_{Q(zeta5)/Q(sqrt5)}(zeta5) = zeta5*zeta5^(-1) = 1. # Norm_{Q(zeta5)/Q(sqrt5)}(1+zeta5) = (1+zeta5)*(1+zeta5^(-1)) = (3+sqrt5)/2, # the square of the fundamental unit (1+sqrt(5))/2 in Z[(1+sqrt(5))/2]. # Norm_{Q(zeta5)/Q(sqrt5)}(2) = 2^2. # Norm_{Q(zeta5)/Q(sqrt5)}(1-zeta5) = (1-zeta5)*(1-zeta5^(-1)) = (5-sqrt5)/2, # which is the same as: sqrt5*((1-sqrt5)/2). # so presumably again we can use "norm is a 5th power in Q(sqrt5)" # to reduce to . # In summary, Q, ring of integers Z, use: <2,5>. # Q(sqrt5), ring of integers Z[(1+sqrt5)/2], use: <(1+sqrt5)/2>. # Q(zeta5), ring of integers Z[zeta5], use: . # Note also that: # This seems to give a naive rank bound of 4, even before local arguments, # since at most 5 indep generators, and subtract 1 for T_2 in J[phi](Q). # Next step is to find: # Jtilde(Q_p)/phi(J(Q_p)) # J(Q_p)/phitilde(Jtilde(Q_p)) # in terms of # J(Q_p)[phi], # Jtilde(Q_p)[phitilde] and |5|_p^2. # Most likely (imitating p.111 of Prolegomena): # # Jtilde(Q_p)/phi(J(Q_p)) # J(Q_p)/phitilde(Jtilde(Q_p)) # = # J[phi](Q_p) # Jtilde[phitilde](Q_p) / |5|_p^2. # I should classify how this looks for different congruence # classes of primes. # # Comment: there are 5^4 5-torsion points over the closure, # and so 5^4 - 5^2 = 5^2*(5^2 - 1) members of J[5] which are # not in the kernel of phi. These are precisely the preimages # under phi of the non-identity members of Jtilde[phitilde]. # It follows that there are no other members of J[5](Q), # apart from , since these are the only members of J[phi](Q), # and any other members of J[5](Q) would map under phi # to non-identity members of Jtilde[phitilde](Q), of which there are none. # # Comment for later. We shall next look at the case r=1. # After this, we might twist by r=p prime, such that: # p == 1 (mod 8), p == 4 (mod 5), so that x |--> x^5 is onto, # and 5 would be a 5th power in Q_p (maybe not needed if that # turns out not to be in Sha). Maybe also want p to satisfy # the the quintic f(x) is irred mod p, as that p (inert) # is added to M, for the 2-descent. Then p would be square # in Q_2,Q_5,R. By the way, note that the class number of the # quintic field for the complete 2-descent has class number 1, # as can be seen using the Magma commands: # _ := PolynomialRing(Rationals()); # K := NumberField( 5*(16 - 12*x + 3*x^2)^2 + (x - 4)^5 ); # ClassNumber(K); # # Look in detail at the case r=1. sqrt5 := RootOf(t^2 - 5); zeta5 := RootOf(t^4 + t^3 + t^2 + t + 1); ssqrt5 := 2*zeta5 + 2*zeta5^(-1) + 1; # Cmon1: y^2 = 5*(16 - 12*x + 3*x^2)^2 + (x - 4)^5 # = (16 - 20*x + 5*x^2)^2 + x^5 Cmon1 := expand( 5*(16 - 12*x + 3*x^2)^2 + (x - 4)^5 ); # T1 = { (4, 16*sqrt5, infty }. # T2 = { (0,16), infty }. # Dmon1: y^2 = 25*(-5+2*sqrt5)*(x^2+(15+4*sqrt5)*100)^2 + (x-10*sqrt5)^5 # = 25*(-5-2*sqrt5)*(x^2+(15-4*sqrt5)*100)^2 + (x+10*sqrt5)^5 Dmon1 := simplify( 25*(-5-2*sqrt5)*(x^2+(15-4*sqrt5)*100)^2 + (x+10*sqrt5)^5 ); # DT1 := {(10*(1+2*zeta5+2*zeta5^4),4000*(2+4*zeta5+zeta5^2+3*zeta5^3),infty} # DT2 := {(10*(-1-2*zeta5-2*zeta5^4),4000*(1+2*zeta5+3*zeta5^2-zeta5^3),infty} # Note that zeta5 |--> zeta5^(-1) (conj of Q(zeta5) over Q(ssqrt5)) # takes DT1 |--> -DT1 and DT2 |--> -DT2, and so has the same effect # on anything in . The only way that something in this group # could be defined over Q(ssqrt5) would be for it to be 2-torsion, # which is only true of the identity in . Hence only the # trivial subgroup of is defined over Q(ssqrt5), with # all nontrivial elements defined over Q(zeta5), but not Q(ssqrt5). # Note also that zeta5 |--> zeta5^2 takes DT1 |--> DT2, # and takes DT2 |--> -DT1, and so takes to , which # shows directly that is defined over Q. # # qtilde : Jac(Cmon1)(Q)/ phitilde( Jac(Dmon1)(Q) ) # --> Q(sqrt(5))*/( Q(sqrt(5))* )^5 x Q*/( Q* )^5 # : {(x1,y1),(x2,y2)} |--> #[(y1-sqrt5*(16-12*x1+3*x1^2))*(y2-sqrt5*(16-12*x2+3*x2^2)), # (y1-(16-20*x1+5*x1^2))*(y2-(16-20*x2+5*x2^2))]. # The image of qtilde is a subset of: [ <(1+sqrt5)/2>, <2,5> ]. # Since (1+sqrt5)/2 = 1/( (1-sqrt5)/2 ) mod 5th powers, # it will turn out to be more convenient to take instead: # The image of qtilde is a subset of: [ <(1-sqrt5)/2>, <2,5> ]. # # q : Jac(Dmon1)(Q)/ phi( Jac(Cmon1)(Q) ) # --> Q(sqrt((-5+2*sqrt5)))*/( Q(sqrt((-5+2*sqrt5)))* )^5 # x Q(sqrt((-5-2*sqrt5)))*/( Q(sqrt((-5-2*sqrt5)))* )^5 # : {(x1,y1),(x2,y2)} |--> # [ (y1 - 5*sqrt((-5+2*sqrt5))*(x1^2 + (15+4*sqrt5)*100)) # *(y2 - 5*sqrt((-5+2*sqrt5))*(x2^2 + (15+4*sqrt5)*100)), # (y1 - 5*sqrt((-5-2*sqrt5))*(x1^2 + (15-4*sqrt5)*100)) # *(y2 - 5*sqrt((-5-2*sqrt5))*(x2^2 + (15-4*sqrt5)*100)) ]. zeta5 := RootOf(t^4 + t^3 + t^2 + t + 1); # The following expresses Q(ssqrt5) explicitly as a subfield of Q(zeta5). ssqrt5 := 2*zeta5 + 2*zeta5^(-1) + 1; factor(x^2 - (-5+2*ssqrt5)); # This gave roots: 1+2*zeta5+2*zeta5^3, -(1+2*zeta5+2*zeta5^3). # Use the following as the Cassels map on the first component. # (y - 5*(1+2*zeta5+2*zeta5^3)*(x^2 + (15+4*(1+2*zeta5+2*zeta5^4))*100)) # The following gives y^2, as we want. simplify( (y - 5*(1+2*zeta5+2*zeta5^3)*(x^2 + (15+4*(1+2*zeta5+2*zeta5^4))*100)) *(y + 5*(1+2*zeta5+2*zeta5^3)*(x^2 + (15+4*(1+2*zeta5+2*zeta5^4))*100)) - (x-10*(1+2*zeta5+2*zeta5^4))^5 + subs(r=1, Dmon) ); factor(x^2 - (-5-2*ssqrt5)); # This gave roots: 1+2*zeta5+2*zeta5^2, -(1+2*zeta5+2*zeta5^2). # Use the following as the Cassels map on the second component. # (y - 5*(1+2*zeta5+2*zeta5^2)*(x^2 + (15-4*(1+2*zeta5+2*zeta5^4))*100)) # The following gives y^2, as we want. simplify( (y - 5*(1+2*zeta5+2*zeta5^2)*(x^2 + (15-4*(1+2*zeta5+2*zeta5^4))*100)) *(y + 5*(1+2*zeta5+2*zeta5^2)*(x^2 + (15-4*(1+2*zeta5+2*zeta5^4))*100)) - (x+10*(1+2*zeta5+2*zeta5^4))^5 + subs(r=1, Dmon) ); # So, the Cassels map in this direction becomes: # q : Jac(Dmon1)(Q)/ phi( Jac(Cmon1)(Q) ) # --> Q(zeta5)*/( Q(zeta5)* )^5 # x Q(zeta5)*/( Q(zeta5)* )^5 # : {(x1,y1),(x2,y2)} |--> # [ (y1 - 5*(1+2*zeta5+2*zeta5^3)*(x1^2 + (15+4*(1+2*zeta5+2*zeta5^4))*100)) # *(y2 - 5*(1+2*zeta5+2*zeta5^3)*(x2^2 + (15+4*(1+2*zeta5+2*zeta5^4))*100)) # (y1 - 5*(1+2*zeta5+2*zeta5^2)*(x1^2 + (15-4*(1+2*zeta5+2*zeta5^4))*100)) # *(y2 - 5*(1+2*zeta5+2*zeta5^2)*(x2^2 + (15-4*(1+2*zeta5+2*zeta5^4))*100)) ] # The image of q is contained in: [ , ]. # But note that zeta5 |--> zeta5^2 takes the first cpt to the second cpt, # so in fact that image of q is contained in: < [zeta, zeta^2] >. # This means our naive bound on the rank (before any local computations) # is actually 3, rather than 4. # # Here are a few comments about the extensions of the relevant Q_p. # For p not dividing n, [Q_p(zeta_n): Q_p] = [F_p(zeta_n) : F_p]. # Note that x^4 + x^3 + x^2 + x + 1 is irred over F_2, # so: [F_2(zeta5) : F_2] = 4, and so: [Q_2(zeta5) : Q_2] = 4, # there is no ramification and the residue field F_2(zeta5) has deg 4 over F_2. # Q_5(zeta5) = Q_5(1-zeta5) and |1-zeta5|_5 = 5^(-1/4). # Here [Q_5(zeta5) : Q_5] = 4, completely ramified. # [Q_2(sqrt5) : Q_2] = 2, no ramification, res field F_2((1-sqrt5)/2) deg 2 # over F_2 (by this, we mean: F_2 adjoin a root of x^2 - x - 1). # Q_5(sqrt5) has |sqrt5|_5 = 5^(-1/2). # Here [Q_5(sqrt5) : Q_5] = 2, completely ramified. # For the finite primes, the relevants algebras remain of the same # degree and do not split down. # Of course R(zeta5) = C. # Also, R(sqrt5) = R, so in this case need to consider the two # embeddings with sqrt5 being positive or negative. # # T2 = { (0,16), infty } in J[phi](Q). qtildeT2 := subs(x=0,y=16,[(y-sqrt5*(16-12*x+3*x^2)), (y-(16-20*x+5*x^2))]); # The above doesn't work, as the second cpt is 0. # Instead use that: (y-(16-20*x+5*x^2))*(y+(16-20*x+5*x^2)) = x^5, # so modulo 5th powers: (y-(16-20*x+5*x^2)) = 1/(y+(16-20*x+5*x^2)). qtildeT2 := subs(x=0,y=16,[(y-sqrt5*(16-12*x+3*x^2)), 1/(y+(16-20*x+5*x^2))]); # This gave: # qtildeT2 := [16-16*RootOf(_Z^2-5), 1/32]. # But 32 is a 5th power, so mod 5th powers, this is just: # qtildeT2 := [(1-sqrt5)/2, 1]. # # We are now ready to consider each of 2,5,infty to get the (5,5)-Selmer bound. # We have the usual commutative diagrams. # Let J denote Jac(Cmon1) and Jtilde denote Jac(Dmon1). # The following literally applies for finite primes (since the extension # degrees over Q_p are the same as over Q). # For infty, should interpret "R(zeta5)" as C and "R(sqrt5)" as R x R, # with positive sqrt5 in first cpt and negative sqrt5 in second cpt. # # # Jtilde(Q)/phi(J(Q)) ---q---> Q(zeta5)*/(Q(zeta5)*)^5 # x Q(zeta5)*/(Q(zeta5)*)^5 # | | # | j | j_p # | | # v v # Jtilde(Q_p)/phi(J(Q_p)) ---q_p---> Q_p(zeta5)*/(Q_p(zeta5)*)^5 # x Q_p(zeta5)*/(Q_p(zeta5)*)^5 # # # J(Q)/phitilde(Jtilde(Q)) ---qtilde-> Q(sqrt5)*/(Q(sqrt5)*)^5 x Q*/(Q*)^5 # | | # | jtilde | jtilde_p # | | # v v # J(Q_p)/phitilde(Jtilde(Q_p)) --qtilde_p--> Q_p(sqrt5)*/(Q_p(sqrt5)*)^5 # x Q_p*/(Q_p*)^5 # # # (Preliminary comment: it looks as if we gain no info at infty, # since R*/(R*)^5 and C*/(C*)^5 are the trivial group. # First consider p=2. # # Jtilde(Q_2)/phi(J(Q_2)) # J(Q_2)/phitilde(Jtilde(Q_2)) # = # J[phi](Q_2) # Jtilde[phitilde](Q_2) / |5|_2^2. # But note that sqrt5, zeta5 are not in Q_2, and T2 is in # J[phi](Q), and so in J[phi](Q_2), so that: # # J[phi](Q_2) = 5^1 and # Jtilde[phitilde](Q_2) = 5^0. Hence: # # Jtilde(Q_2)/phi(J(Q_2)) # J(Q_2)/phitilde(Jtilde(Q_2)) = 5^1. # First find the kernel of j_2. # Want to know whether or not zeta5 is in (Q_2(zeta5)*)^5. # Imagine x in Q_2(zeta5) satisfied x^5 = zeta5. # Then x^20+x^15+x^10+x^5+1 = 0. However: Factor(x^20+x^15+x^10+x^5+1) mod 2; # shows that this is irred mod 2 and so irred in Q_2, and this # would mean a degree 20 extension of Q_2 contained inside Q_2(zeta5), # which is a degree 4 extension of Q_2, a contradiction. # Hence j_2 has trivial kernel: [ <1>, <1> ] = < [1,1] >. # # Now find the kernel of jtilde_2. # First see whether or not (1-sqrt5)/2 is in the kernel of the first # component of jtilde_2. That is, whether (1-sqrt5)/2 is in (Q_2(sqrt5)*)^5. # Want: ( (x + y*sqrt5)/2 )^5 - (1 - sqrt5)/2 = 0. ww := simplify( (x + y*sqrt5)^5 - 2^4*(1 - sqrt5) ); # We want to know if there are x,y in Q_2 satisfying the above. eq1 := coeff(ww,sqrt5); eq2 := simplify( ww - eq1*sqrt5 ); # We need eq1,eq2 (in x,y) to have a common soln: zz := subs(factor(resultant(eq1,eq2,y))); # This has a factor: x^5+5*x^3+5*x-1, which clearly has a solution # in Q_2, by replacing x by 1/x, to solve: 1+5*x^2+5*x^4-x^5 # and then use x=1 and HL to see that this has a soln in Q_2. # This only show that there is a value of x for which eq1,eq2 # have a common factor in y. However, note also that: simplify( ( (-1 - sqrt5)/2 )^5 - (1 - sqrt5)/2 ); # gives: -6 - 2*sqrt5, so we not that if: f(x) = x^5 - (1 - sqrt5)/2 # then | f( (-1 - sqrt5)/2 ) |_2 < 1, and: # | f'( (-1 - sqrt5)/2 ) |_2 = | 5*( (-1 - sqrt5)/2 )^4 |_2 = 1, # so by HL (1-sqrt5)/2 is in (Q_2(sqrt5)*)^5. # Hence (1-sqrt5)/2 is in the kernel of jtilde_2. # Also, x^5 - 5 has a soln in Q_2 by x=1 and HL. # However, x^5 - 2 does not have a soln in Q_2, using valuations. # So, the kernel of jtilde_2 is precisely: [ <(1-sqrt5)/2>, <5> ]. with(padic); for xx from -17 to 17 do if nops([rootp(y^2 - subs(x=xx*2, Cmon1), 2)]) > 0 then print(xx*2, ifactor(subs(x=xx*2, Cmon1))) fi od; # The following is the second component of the map qtilde_2. qq := y-(16-20*x+5*x^2); [msolve( y^2 - subs(x=-8, Cmon1), 2^30 )][1][1]; ifactor(subs( x=-8, %, qq)); # The above didn't give anything non-identity, so likely we already # know all of the image of qtilde_2. for xx from -33 to 33 do if nops([rootp(y^2 - subs(x=xx, Dmon1), 2)]) > 0 then print(xx, ifactor(subs(x=xx, Dmon1))) fi od; # The following is the first cpt of the map q_2. rr := (y - 5*(1+2*zeta5+2*zeta5^3)*(x^2 + (15+4*(1+2*zeta5+2*zeta5^4))*100)); # The following is the second cpt of the map q_2. rrr := (y - 5*(1+2*zeta5+2*zeta5^2)*(x^2 + (15-4*(1+2*zeta5+2*zeta5^4))*100)); [msolve( y^2 - subs(x=-2, Dmon1), 2^50 )][1][1]; simplify(subs( x=-2, %, rr)); simplify(subs( x=-2, y=758703353981424, rr)); # In the above, we selected y == 48 (mod 64), rather than y == 16 (mod 64). # # Now have to see whether the above is zeta5*(something in (Q_2(zeta5)*)^5). simplify(%/zeta5)/32; % mod 2; # The above gave: zeta5^2 + zeta5^3. simplify(subs( x=-2, y=758703353981424, rrr)); simplify(%/zeta5^2)/32; % mod 2; # The above gave: 1 + zeta5^2 + zeta5^3. # for i1 from 0 to 1 do for i2 from 0 to 1 do for i3 from 0 to 1 do # for i4 from 0 to 1 do # print( simplify( (i1 + i2*zeta5 + i3*zeta5^2 + i4*zeta5^3)^5 ) mod 2 ) # od od od od; # The above gives that the following are the 5th powers mod 2: # 1, zeta5^2 + zeta5^3, 1 + zeta5^2 + zeta5^3. # In particular, (1+zeta5^2)^5 = -8 - 5*zeta5^2 - 5*zeta5^3, # which is congruent to: zeta5^2 + zeta5^3 mod 2. # So, take y in Q_2 to be such that y == 48 (mod 64) and # satisfying: y^2 = subs(x=-2, Dmon1) = -2^8*330047 (exists, since # -330047 == 1 mod 8). Then { (x,y), infty } is in Jtilde(Q_2) # and maps under q_2 to [zeta5*k, zeta5^2*l] where k is in Q_2(zeta5) # and k == zeta5^2 + zeta5^3 mod 2. Then | (1+zeta5^2)^5 - k |_2 < 1 and # | 5(1+zeta5^2)^2 |_2 = 1, so apply HL to see that k is in (Q_2(zeta5)*)^5. # Similarly l is in (Q_2(zeta5)*)^5. # Hence q_2 : { (x,y), infty } |--> [zeta5, zeta5^2]. # So not we see that we have all of: # # Jtilde(Q_2)/phi(J(Q_2)) = 5^1, with image < [zeta5,zeta5^2] >. # # J(Q_2)/phitilde(Jtilde(Q_2)) = 5^0. # Also, j_2 has trivial kernel and # the kernel of jtilde_2 is precisely: [ <(1-sqrt5)/2>, <5> ], # so that: # j_2^{-1}( q_2( Jtilde(Q_2)/phi(J(Q_2)) ) ) = < [zeta5,zeta5^2] >. # jtilde_2^{-1}( qtilde_2( J(Q_2)/phitilde(Jtilde(Q_2)) ) ) # = [ <(1-sqrt5)/2>, <5> ] = < [(1-sqrt5)/2,1], [1,5] >. # At this point, the rank of J(Q) is bounded above by 2. # # Now consider p=5. # # Jtilde(Q_5)/phi(J(Q_5)) # J(Q_5)/phitilde(Jtilde(Q_5)) # = # J[phi](Q_5) # Jtilde[phitilde](Q_5) / |5|_5^2. # But note that sqrt5, zeta5 are not in Q_5, and T2 is in # J[phi](Q), and so in J[phi](Q_5), so that: # # J[phi](Q_5) = 5^1 and # Jtilde[phitilde](Q_5) = 5^0. Hence: # # Jtilde(Q_5)/phi(J(Q_5)) # J(Q_5)/phitilde(Jtilde(Q_5)) = 5^3. # First find the kernel of j_5. # Want to know whether or not zeta5 is in (Q_5(zeta5)*)^5. # Imagine x in Q_5(zeta5) satisfied x^5 = zeta5. # Then: |x|_5 = 1. Also: (x - 1)^5 == x^5 - 1 == zeta5 - 1 (mod 5), # and |zeta5 - 1|_5 = 5^(-1/5), but that would force |x-1|_5 = 5^(-1/25) # which is too much ramification for Q_5(zeta_5). # Hence j_5 has trivial kernel: [ <1>, <1> ] = < [1,1] >. # # Now find the kernel of jtilde_5. # First see whether or not (1-sqrt5)/2 is in the kernel of the first # component of jtilde_5. That is, whether (1-sqrt5)/2 is in (Q_5(sqrt5)*)^5. ww := simplify( 2*(x + y*sqrt5)^5 - (1 - sqrt5) ); # We want to know if there are x,y in Q_5 satisfying the above. eq1 := coeff(ww,sqrt5); eq2 := simplify( ww - eq1*sqrt5 ); # We need eq1,eq2 (in x,y) to have a common soln: zz := subs(x=x/2,factor(resultant(eq1,eq2,y))); rootp(zz/6250000,5); # This has no solutions, so no value of x for which eq1,eq2 has # common factor in y, much less a common soln. # This can alternatively be seen by noting that: # for i1 from 0 to 4 do for i2 from 0 to 4 do # print( simplify( (i1 + i2*sqrt5)^5 - (1 - sqrt5)/2 ) ) od od; # never gives anything congruent to 0 (mod 5), so that # x^5 == (1 - sqrt5)/2 has no solutions mod 5. Indeed, this # is easy to see directly since, after expanding (i1 + i2*sqrt5)^5 # we get i1 (mod 5) which can never be == (1 - sqrt5)/2 (mod 5). # Hence (1-sqrt5)/2 is not in the kernel of jtilde_5. # Also, rootp(x^5-5,2) shows x^5 - 2 has no soln in Q_5, # and x^5 = 5^i*2^j, for 5 not dividing i, does not have a soln in Q_5, # using valuations. # So, jtilde_5 has trivial kernel. # # First recall that T2 = { (0,16), infty } in J[phi](Q) and so in J[phi](Q_5). # qtildeT2 := [(1-sqrt5)/2, 1]. # So qtilde_2 : T2 |--> [(1-sqrt5)/2, 1], and this is nontrivial, # since it is not in the kernel of jtilde_5. # We still require two further generators. for xx from -17 to 17 do if nops([rootp(y^2 - subs(x=xx, Cmon1), 5)]) > 0 then print(xx, ifactor(subs(x=xx, Cmon1))) fi od; # The following is the second component of the map qtilde_5. qq := y-(16-20*x+5*x^2); [msolve( y^2 - subs(x=3, Cmon1), 5^8 )][1][1]; # The above gave: y = 257412 rootp( w^5 - subs( x=3, y=257412, qq)/2, 5 ); # This has a solution, so this is just 2, mod 5th powers in Q_5, # so the image of qtilde_5 contains [ ?, 2 ], and so contains # [1,2], since we already know that [(1-sqrt5)/2, 1] is in the image. # Note that something might map to an element of Q_5*/(Q_5*)^5, # which is not even in the image of the map jtilde_5, # so that would potentially give rank 0. # Might describe completely: Q_5*/(Q_5*)^5. # In fact, all points seem to give elements of <2>, so need to # check q_5, and it might be q_5 for which something might # map to an element of Q_5(zeta5)*/(Q_5(zeta5)*)^5 which # is not even in the image of the map j_5. # for xx from -33 to 33 do if nops([rootp(y^2 - subs(x=xx, Dmon1), 5)]) > 0 then print(xx, ifactor(subs(x=xx, Dmon1))) fi od; # The following is the first cpt of the map q_5. rr := (y - 5*(1+2*zeta5+2*zeta5^3)*(x^2 + (15+4*(1+2*zeta5+2*zeta5^4))*100)); # The following is the second cpt of the map q_5. rrr := (y - 5*(1+2*zeta5+2*zeta5^2)*(x^2 + (15-4*(1+2*zeta5+2*zeta5^4))*100)); [msolve( y^2 - subs(x=1, Dmon1), 5^8 )][1][1]; simplify(subs( x=1, %, rr)); simplify(subs( x=-2, y=758703353981424, rr)); # Not yet found the generator on this side. # Most likely: should find something d for which: # d, d*zeta5, d*zeta5^2, d*zeta5^3, d*zeta5^4 are not 5th powers in Q_5(zeta5). # I should first describe precisely the 5th powers mod some power of 5. # Maybe also describe completely: Q_5(zeta5)*/(Q_5(zeta5)*)^5. # # arr := {}; # for i1 from 0 to 24 do print(i1); # for i2 from 0 to 24 do for i3 from 0 to 24 do for i4 from 0 to 24 do # arr := { op(arr), # simplify( (i1 + i2*zeta5 + i3*zeta5^2 + i4*zeta5^3)^5 ) mod 5^3 }; # od od od od; # arr4 := [ op(arr) ]; # The array arr4 is enormous and is in the file: ed/arr4 # We shall reduce this mod 5^2. # # arr5a := []; # for i from 1 to nops(arr4) do arr5a := [ op(arr5a), arr4[i] mod 5^2 ] od: # arr5a := [ op( { op(arr5a) } ) ]; # arr6a := []; # for i from 1 to nops(arr5a) do # arr6a := [ op(arr6a), subs(zeta5 = zzet, arr5a[i]) ] od; # for i from 1 to nops(arr6a) do print( arr6a[i], " " ) od; # # arr5b := []; # for i from 1 to nops(arr4) do # arr5b := [ op(arr5b), expand(simplify(zeta5*arr4[i])) mod 5^2 ] od: # arr5b := [ op( { op(arr5b) } ) ]; # arr6b := []; # for i from 1 to nops(arr5b) do # arr6b := [ op(arr6b), subs(zeta5 = zzet, arr5b[i]) ] od; # for i from 1 to nops(arr6b) do print( arr6b[i], " " ) od; # # arr5c := []; # for i from 1 to nops(arr4) do # arr5c := [ op(arr5c), expand(simplify(zeta5^2*arr4[i])) mod 5^2 ] od: # arr5c := [ op( { op(arr5c) } ) ]; # arr6c := []; # for i from 1 to nops(arr5c) do # arr6c := [ op(arr6c), subs(zeta5 = zzet, arr5c[i]) ] od; # for i from 1 to nops(arr6c) do print( arr6c[i], " " ) od; # # arr5d := []; # for i from 1 to nops(arr4) do # arr5d := [ op(arr5d), expand(simplify(zeta5^3*arr4[i])) mod 5^2 ] od: # arr5d := [ op( { op(arr5d) } ) ]; # arr6d := []; # for i from 1 to nops(arr5d) do # arr6d := [ op(arr6d), subs(zeta5 = zzet, arr5d[i]) ] od; # for i from 1 to nops(arr6d) do print( arr6d[i], " " ) od; # # arr5e := []; # for i from 1 to nops(arr4) do # arr5e := [ op(arr5e), expand(simplify(zeta5^4*arr4[i])) mod 5^2 ] od: # arr5e := [ op( { op(arr5e) } ) ]; # arr6e := []; # for i from 1 to nops(arr5e) do # arr6e := [ op(arr6e), subs(zeta5 = zzet, arr5e[i]) ] od; # for i from 1 to nops(arr6e) do print( arr6e[i], " " ) od; # # The following gives the members of (Q_5(zeta)*)^2 mod 5^2. arr6a := [ 0, 1, 7, 18, 24, 5*zeta5^2+15*zeta5+4, 5*zeta5^2+15*zeta5+6, 5*zeta5^2+15*zeta5+12, 5*zeta5^2+15*zeta5+23, 10*zeta5^2+5*zeta5+3, 10*zeta5^2+5*zeta5+9, 10*zeta5^2+5*zeta5+11, 10*zeta5^2+5*zeta5+17, 15*zeta5^2+20*zeta5+8, 15*zeta5^2+20*zeta5+14, 15*zeta5^2+20*zeta5+16, 15*zeta5^2+20*zeta5+22, 20*zeta5^2+10*zeta5+2, 20*zeta5^2+10*zeta5+13, 20*zeta5^2+10*zeta5+19, 20*zeta5^2+10*zeta5+21, 5*zeta5^3+5*zeta5^2+8, 5*zeta5^3+5*zeta5^2+14, 5*zeta5^3+5*zeta5^2+16, 5*zeta5^3+5*zeta5^2+22, 5*zeta5^3+10*zeta5+3, 5*zeta5^3+10*zeta5+9, 5*zeta5^3+10*zeta5+11, 5*zeta5^3+10*zeta5+17, 10*zeta5^3+10*zeta5^2+4, 10*zeta5^3+10*zeta5^2+6, 10*zeta5^3+10*zeta5^2+12, 10*zeta5^3+10*zeta5^2+23, 10*zeta5^3+20*zeta5+2, 10*zeta5^3+20*zeta5+13, 10*zeta5^3+20*zeta5+19, 10*zeta5^3+20*zeta5+21, 15*zeta5^3+15*zeta5^2+2, 15*zeta5^3+15*zeta5^2+13, 15*zeta5^3+15*zeta5^2+19, 15*zeta5^3+15*zeta5^2+21, 15*zeta5^3+5*zeta5+4, 15*zeta5^3+5*zeta5+6, 15*zeta5^3+5*zeta5+12, 15*zeta5^3+5*zeta5+23, 20*zeta5^3+20*zeta5^2+3, 20*zeta5^3+20*zeta5^2+9, 20*zeta5^3+20*zeta5^2+11, 20*zeta5^3+20*zeta5^2+17, 20*zeta5^3+15*zeta5+8, 20*zeta5^3+15*zeta5+14, 20*zeta5^3+15*zeta5+16, 20*zeta5^3+15*zeta5+22, 5*zeta5^3+10*zeta5^2+15*zeta5+2, 5*zeta5^3+10*zeta5^2+15*zeta5+13, 5*zeta5^3+10*zeta5^2+15*zeta5+19, 5*zeta5^3+10*zeta5^2+15*zeta5+21, 5*zeta5^3+15*zeta5^2+5*zeta5+1, 5*zeta5^3+15*zeta5^2+5*zeta5+7, 5*zeta5^3+15*zeta5^2+5*zeta5+18, 5*zeta5^3+15*zeta5^2+5*zeta5+24, 5*zeta5^3+15*zeta5^2+20*zeta5+10, 5*zeta5^3+20*zeta5^2+20*zeta5+4, 5*zeta5^3+20*zeta5^2+20*zeta5+6, 5*zeta5^3+20*zeta5^2+20*zeta5+12, 5*zeta5^3+20*zeta5^2+20*zeta5+23, 10*zeta5^3+5*zeta5^2+10*zeta5+1, 10*zeta5^3+5*zeta5^2+10*zeta5+7, 10*zeta5^3+5*zeta5^2+10*zeta5+18, 10*zeta5^3+5*zeta5^2+10*zeta5+24, 10*zeta5^3+5*zeta5^2+15*zeta5+20, 10*zeta5^3+15*zeta5^2+15*zeta5+3, 10*zeta5^3+15*zeta5^2+15*zeta5+9, 10*zeta5^3+15*zeta5^2+15*zeta5+11, 10*zeta5^3+15*zeta5^2+15*zeta5+17, 10*zeta5^3+20*zeta5^2+5*zeta5+8, 10*zeta5^3+20*zeta5^2+5*zeta5+14, 10*zeta5^3+20*zeta5^2+5*zeta5+16, 10*zeta5^3+20*zeta5^2+5*zeta5+22, 15*zeta5^3+5*zeta5^2+20*zeta5+3, 15*zeta5^3+5*zeta5^2+20*zeta5+9, 15*zeta5^3+5*zeta5^2+20*zeta5+11, 15*zeta5^3+5*zeta5^2+20*zeta5+17, 15*zeta5^3+10*zeta5^2+10*zeta5+8, 15*zeta5^3+10*zeta5^2+10*zeta5+14, 15*zeta5^3+10*zeta5^2+10*zeta5+16, 15*zeta5^3+10*zeta5^2+10*zeta5+22, 15*zeta5^3+20*zeta5^2+10*zeta5+5, 15*zeta5^3+20*zeta5^2+15*zeta5+1, 15*zeta5^3+20*zeta5^2+15*zeta5+7, 15*zeta5^3+20*zeta5^2+15*zeta5+18, 15*zeta5^3+20*zeta5^2+15*zeta5+24, 20*zeta5^3+5*zeta5^2+5*zeta5+2, 20*zeta5^3+5*zeta5^2+5*zeta5+13, 20*zeta5^3+5*zeta5^2+5*zeta5+19, 20*zeta5^3+5*zeta5^2+5*zeta5+21, 20*zeta5^3+10*zeta5^2+5*zeta5+15, 20*zeta5^3+10*zeta5^2+20*zeta5+1, 20*zeta5^3+10*zeta5^2+20*zeta5+7, 20*zeta5^3+10*zeta5^2+20*zeta5+18, 20*zeta5^3+10*zeta5^2+20*zeta5+24, 20*zeta5^3+15*zeta5^2+10*zeta5+4, 20*zeta5^3+15*zeta5^2+10*zeta5+6, 20*zeta5^3+15*zeta5^2+10*zeta5+12, 20*zeta5^3+15*zeta5^2+10*zeta5+23 ]; # The following gives the members of zeta*(Q_5(zeta)*)^2 mod 5^2. arr6b := [ 0, 7*zeta5, 18*zeta5, 24*zeta5, 5*zeta5^3+15*zeta5^2+4*zeta5, 5*zeta5^3+15*zeta5^2+6*zeta5, 5*zeta5^3+15*zeta5^2+12*zeta5, 5*zeta5^3+15*zeta5^2+23*zeta5, 5*zeta5^3+3*zeta5+10, 5*zeta5^3+9*zeta5+10, 5*zeta5^3+11*zeta5+10, 5*zeta5^3+17*zeta5+10, 10*zeta5^3+5*zeta5^2+3*zeta5, 10*zeta5^3+5*zeta5^2+9*zeta5, 10*zeta5^3+5*zeta5^2+11*zeta5, 10*zeta5^3+5*zeta5^2+17*zeta5, 10*zeta5^3+30*zeta5^2+28*zeta5, 10*zeta5^3+30*zeta5^2+34*zeta5, 10*zeta5^3+30*zeta5^2+36*zeta5, 10*zeta5^3+30*zeta5^2+42*zeta5, 10*zeta5^3+2*zeta5+20, 10*zeta5^3+13*zeta5+20, 10*zeta5^3+19*zeta5+20, 10*zeta5^3+21*zeta5+20, 15*zeta5^3+20*zeta5^2+8*zeta5, 15*zeta5^3+20*zeta5^2+14*zeta5, 15*zeta5^3+20*zeta5^2+16*zeta5, 15*zeta5^3+20*zeta5^2+22*zeta5, 15*zeta5^3+45*zeta5^2+33*zeta5, 15*zeta5^3+45*zeta5^2+39*zeta5, 15*zeta5^3+45*zeta5^2+66*zeta5, 15*zeta5^3+45*zeta5^2+72*zeta5, 15*zeta5^3+4*zeta5+5, 15*zeta5^3+6*zeta5+5, 15*zeta5^3+12*zeta5+5, 15*zeta5^3+23*zeta5+5, 20*zeta5^3+10*zeta5^2+2*zeta5, 20*zeta5^3+10*zeta5^2+13*zeta5, 20*zeta5^3+10*zeta5^2+19*zeta5, 20*zeta5^3+10*zeta5^2+21*zeta5, 20*zeta5^3+10*zeta5^2+38*zeta5, 20*zeta5^3+10*zeta5^2+44*zeta5, 20*zeta5^3+10*zeta5^2+46*zeta5, 20*zeta5^3+60*zeta5^2+44*zeta5, 20*zeta5^3+60*zeta5^2+52*zeta5, 20*zeta5^3+60*zeta5^2+88*zeta5, 20*zeta5^3+60*zeta5^2+96*zeta5, 20*zeta5^3+8*zeta5+15, 20*zeta5^3+14*zeta5+15, 20*zeta5^3+16*zeta5+15, 20*zeta5^3+22*zeta5+15, 30*zeta5^3+15*zeta5^2+6*zeta5, 30*zeta5^3+15*zeta5^2+12*zeta5, 30*zeta5^3+15*zeta5^2+48*zeta5, 30*zeta5^3+15*zeta5^2+54*zeta5, 30*zeta5^3+40*zeta5^2+4*zeta5, 30*zeta5^3+40*zeta5^2+6*zeta5, 30*zeta5^3+40*zeta5^2+12*zeta5, 30*zeta5^3+40*zeta5^2+48*zeta5, 30*zeta5^3+90*zeta5^2+6*zeta5, 30*zeta5^3+90*zeta5^2+12*zeta5, 30*zeta5^3+90*zeta5^2+48*zeta5, 30*zeta5^3+90*zeta5^2+54*zeta5, 35*zeta5^3+105*zeta5^2+28*zeta5, 35*zeta5^3+105*zeta5^2+42*zeta5, 35*zeta5^3+105*zeta5^2+84*zeta5, 40*zeta5^3+20*zeta5^2+8*zeta5, 40*zeta5^3+20*zeta5^2+14*zeta5, 40*zeta5^3+20*zeta5^2+16*zeta5, 40*zeta5^3+20*zeta5^2+22*zeta5, 40*zeta5^3+20*zeta5^2+64*zeta5, 40*zeta5^3+20*zeta5^2+72*zeta5, 40*zeta5^3+120*zeta5^2+8*zeta5, 40*zeta5^3+120*zeta5^2+16*zeta5, 40*zeta5^3+120*zeta5^2+64*zeta5, 40*zeta5^3+120*zeta5^2+72*zeta5, 45*zeta5^3+60*zeta5^2+21*zeta5, 45*zeta5^3+60*zeta5^2+27*zeta5, 45*zeta5^3+60*zeta5^2+63*zeta5, 45*zeta5^3+60*zeta5^2+69*zeta5, 60*zeta5^3+30*zeta5^2+3*zeta5, 60*zeta5^3+30*zeta5^2+9*zeta5, 60*zeta5^3+30*zeta5^2+36*zeta5, 60*zeta5^3+30*zeta5^2+42*zeta5, 60*zeta5^3+30*zeta5^2+78*zeta5, 60*zeta5^3+30*zeta5^2+84*zeta5, 60*zeta5^3+80*zeta5^2+28*zeta5, 60*zeta5^3+80*zeta5^2+36*zeta5, 60*zeta5^3+80*zeta5^2+84*zeta5, 60*zeta5^3+80*zeta5^2+92*zeta5, 70*zeta5^3+35*zeta5^2+21*zeta5, 70*zeta5^3+35*zeta5^2+63*zeta5, 70*zeta5^3+35*zeta5^2+77*zeta5, 70*zeta5^3+35*zeta5^2+119*zeta5, 80*zeta5^3+40*zeta5^2+4*zeta5, 80*zeta5^3+40*zeta5^2+12*zeta5, 80*zeta5^3+40*zeta5^2+48*zeta5, 80*zeta5^3+40*zeta5^2+56*zeta5, 80*zeta5^3+40*zeta5^2+104*zeta5, 80*zeta5^3+40*zeta5^2+112*zeta5, 90*zeta5^3+45*zeta5^2+72*zeta5, 90*zeta5^3+45*zeta5^2+108*zeta5, 90*zeta5^3+120*zeta5^2+66*zeta5, 90*zeta5^3+120*zeta5^2+72*zeta5, 90*zeta5^3+120*zeta5^2+108*zeta5, 90*zeta5^3+120*zeta5^2+114*zeta5, 110*zeta5^3+55*zeta5^2+11*zeta5, 120*zeta5^3+60*zeta5^2+96*zeta5, 120*zeta5^3+60*zeta5^2+102*zeta5, 2*zeta5+15+15*zeta5^2, 3*zeta5+20+20*zeta5^2, 4*zeta5+10+10*zeta5^2, 6*zeta5+10+10*zeta5^2, 8*zeta5+5+5*zeta5^2, 9*zeta5+20+20*zeta5^2, 11*zeta5+20+20*zeta5^2, 12*zeta5+10+10*zeta5^2, 13*zeta5+15+15*zeta5^2, 14*zeta5+5+5*zeta5^2, 16*zeta5+5+5*zeta5^2, 17*zeta5+20+20*zeta5^2, 19*zeta5+15+15*zeta5^2, 21*zeta5+15+15*zeta5^2, 22*zeta5+5+5*zeta5^2, 23*zeta5+10+10*zeta5^2, 5*zeta5^2+4*zeta5+20+20*zeta5^3, 5*zeta5^2+6*zeta5+20+20*zeta5^3, 5*zeta5^2+12*zeta5+20+20*zeta5^3, 5*zeta5^2+23*zeta5+20+20*zeta5^3, 10*zeta5^2+3*zeta5+15+15*zeta5^3, 10*zeta5^2+9*zeta5+15+15*zeta5^3, 10*zeta5^2+11*zeta5+15+15*zeta5^3, 10*zeta5^2+17*zeta5+15+15*zeta5^3, 15*zeta5^2+8*zeta5+10+10*zeta5^3, 15*zeta5^2+14*zeta5+10+10*zeta5^3, 15*zeta5^2+16*zeta5+10+10*zeta5^3, 15*zeta5^2+22*zeta5+10+10*zeta5^3, 20*zeta5^2+2*zeta5+5+5*zeta5^3, 20*zeta5^2+13*zeta5+5+5*zeta5^3, 20*zeta5^2+19*zeta5+5+5*zeta5^3, 20*zeta5^2+21*zeta5+5+5*zeta5^3, 5*zeta5^3+5*zeta5^2+zeta5+15, 5*zeta5^3+5*zeta5^2+7*zeta5+15, 5*zeta5^3+5*zeta5^2+18*zeta5+15, 5*zeta5^3+5*zeta5^2+24*zeta5+15, 5*zeta5^3+10*zeta5^2+8*zeta5+20, 5*zeta5^3+10*zeta5^2+14*zeta5+20, 5*zeta5^3+10*zeta5^2+16*zeta5+20, 5*zeta5^3+10*zeta5^2+22*zeta5+20, 5*zeta5^3+20*zeta5^2+15*zeta5+10, 10*zeta5^3+10*zeta5^2+zeta5+5, 10*zeta5^3+10*zeta5^2+7*zeta5+5, 10*zeta5^3+10*zeta5^2+18*zeta5+5, 10*zeta5^3+10*zeta5^2+24*zeta5+5, 10*zeta5^3+15*zeta5^2+5*zeta5+20, 10*zeta5^3+20*zeta5^2+4*zeta5+15, 10*zeta5^3+20*zeta5^2+6*zeta5+15, 10*zeta5^3+20*zeta5^2+12*zeta5+15, 10*zeta5^3+20*zeta5^2+23*zeta5+15, 15*zeta5^3+5*zeta5^2+2*zeta5+10, 15*zeta5^3+5*zeta5^2+13*zeta5+10, 15*zeta5^3+5*zeta5^2+19*zeta5+10, 15*zeta5^3+5*zeta5^2+21*zeta5+10, 15*zeta5^3+10*zeta5^2+20*zeta5+5, 15*zeta5^3+15*zeta5^2+zeta5+20, 15*zeta5^3+15*zeta5^2+7*zeta5+20, 15*zeta5^3+15*zeta5^2+18*zeta5+20, 15*zeta5^3+15*zeta5^2+24*zeta5+20, 20*zeta5^3+5*zeta5^2+10*zeta5+15, 20*zeta5^3+15*zeta5^2+3*zeta5+5, 20*zeta5^3+15*zeta5^2+9*zeta5+5, 20*zeta5^3+15*zeta5^2+11*zeta5+5, 20*zeta5^3+15*zeta5^2+17*zeta5+5, 20*zeta5^3+20*zeta5^2+zeta5+10, 20*zeta5^3+20*zeta5^2+7*zeta5+10, 20*zeta5^3+20*zeta5^2+18*zeta5+10, 20*zeta5^3+20*zeta5^2+24*zeta5+10, zeta5 ]; # The following gives the members of zeta^2*(Q_5(zeta)*)^2 mod 5^2. arr6c := [ 0, zeta5^2, 7*zeta5^2, 18*zeta5^2, 24*zeta5^2, 2*zeta5^2+10*zeta5+20, 2*zeta5^2+15*zeta5+15*zeta5^3, 3*zeta5^2+5*zeta5+10, 3*zeta5^2+20*zeta5+20*zeta5^3, 4*zeta5^2+10*zeta5+10*zeta5^3, 4*zeta5^2+15*zeta5+5, 6*zeta5^2+10*zeta5+10*zeta5^3, 6*zeta5^2+15*zeta5+5, 8*zeta5^2+5*zeta5+5*zeta5^3, 8*zeta5^2+20*zeta5+15, 9*zeta5^2+5*zeta5+10, 9*zeta5^2+20*zeta5+20*zeta5^3, 11*zeta5^2+5*zeta5+10, 11*zeta5^2+20*zeta5+20*zeta5^3, 12*zeta5^2+10*zeta5+10*zeta5^3, 12*zeta5^2+15*zeta5+5, 13*zeta5^2+10*zeta5+20, 13*zeta5^2+15*zeta5+15*zeta5^3, 14*zeta5^2+5*zeta5+5*zeta5^3, 14*zeta5^2+20*zeta5+15, 16*zeta5^2+5*zeta5+5*zeta5^3, 16*zeta5^2+20*zeta5+15, 17*zeta5^2+5*zeta5+10, 17*zeta5^2+20*zeta5+20*zeta5^3, 19*zeta5^2+10*zeta5+20, 19*zeta5^2+15*zeta5+15*zeta5^3, 21*zeta5^2+10*zeta5+20, 21*zeta5^2+15*zeta5+15*zeta5^3, 22*zeta5^2+5*zeta5+5*zeta5^3, 22*zeta5^2+20*zeta5+15, 23*zeta5^2+10*zeta5+10*zeta5^3, 23*zeta5^2+15*zeta5+5, 5*zeta5^3+4*zeta5^2+15, 5*zeta5^3+6*zeta5^2+15, 5*zeta5^3+12*zeta5^2+15, 5*zeta5^3+23*zeta5^2+15, 10*zeta5^3+3*zeta5^2+5, 10*zeta5^3+9*zeta5^2+5, 10*zeta5^3+11*zeta5^2+5, 10*zeta5^3+17*zeta5^2+5, 15*zeta5^3+8*zeta5^2+20, 15*zeta5^3+14*zeta5^2+20, 15*zeta5^3+16*zeta5^2+20, 15*zeta5^3+22*zeta5^2+20, 20*zeta5^3+2*zeta5^2+10, 20*zeta5^3+13*zeta5^2+10, 20*zeta5^3+19*zeta5^2+10, 20*zeta5^3+21*zeta5^2+10, 2*zeta5^2+20*zeta5+5+5*zeta5^3, 3*zeta5^2+10*zeta5+15+15*zeta5^3, 4*zeta5^2+5*zeta5+20+20*zeta5^3, 6*zeta5^2+5*zeta5+20+20*zeta5^3, 8*zeta5^2+15*zeta5+10+10*zeta5^3, 9*zeta5^2+10*zeta5+15+15*zeta5^3, 11*zeta5^2+10*zeta5+15+15*zeta5^3, 12*zeta5^2+5*zeta5+20+20*zeta5^3, 13*zeta5^2+20*zeta5+5+5*zeta5^3, 14*zeta5^2+15*zeta5+10+10*zeta5^3, 16*zeta5^2+15*zeta5+10+10*zeta5^3, 17*zeta5^2+10*zeta5+15+15*zeta5^3, 19*zeta5^2+20*zeta5+5+5*zeta5^3, 21*zeta5^2+20*zeta5+5+5*zeta5^3, 22*zeta5^2+15*zeta5+10+10*zeta5^3, 23*zeta5^2+5*zeta5+20+20*zeta5^3, 5*zeta5^3+zeta5^2+10+10*zeta5, 5*zeta5^3+3*zeta5^2+15*zeta5+20, 5*zeta5^3+7*zeta5^2+10+10*zeta5, 5*zeta5^3+9*zeta5^2+15*zeta5+20, 5*zeta5^3+11*zeta5^2+15*zeta5+20, 5*zeta5^3+17*zeta5^2+15*zeta5+20, 5*zeta5^3+18*zeta5^2+10+10*zeta5, 5*zeta5^3+20*zeta5^2+10*zeta5+15, 5*zeta5^3+24*zeta5^2+10+10*zeta5, 10*zeta5^3+zeta5^2+20+20*zeta5, 10*zeta5^3+2*zeta5^2+5*zeta5+15, 10*zeta5^3+7*zeta5^2+20+20*zeta5, 10*zeta5^3+13*zeta5^2+5*zeta5+15, 10*zeta5^3+15*zeta5^2+20*zeta5+5, 10*zeta5^3+18*zeta5^2+20+20*zeta5, 10*zeta5^3+19*zeta5^2+5*zeta5+15, 10*zeta5^3+21*zeta5^2+5*zeta5+15, 10*zeta5^3+24*zeta5^2+20+20*zeta5, 15*zeta5^3+zeta5^2+5+5*zeta5, 15*zeta5^3+4*zeta5^2+20*zeta5+10, 15*zeta5^3+6*zeta5^2+20*zeta5+10, 15*zeta5^3+7*zeta5^2+5+5*zeta5, 15*zeta5^3+10*zeta5^2+5*zeta5+20, 15*zeta5^3+12*zeta5^2+20*zeta5+10, 15*zeta5^3+18*zeta5^2+5+5*zeta5, 15*zeta5^3+23*zeta5^2+20*zeta5+10, 15*zeta5^3+24*zeta5^2+5+5*zeta5, 20*zeta5^3+zeta5^2+15+15*zeta5, 20*zeta5^3+5*zeta5^2+15*zeta5+10, 20*zeta5^3+7*zeta5^2+15+15*zeta5, 20*zeta5^3+8*zeta5^2+10*zeta5+5, 20*zeta5^3+14*zeta5^2+10*zeta5+5, 20*zeta5^3+16*zeta5^2+10*zeta5+5, 20*zeta5^3+18*zeta5^2+15+15*zeta5, 20*zeta5^3+22*zeta5^2+10*zeta5+5, 20*zeta5^3+24*zeta5^2+15+15*zeta5 ]; # The following gives the members of zeta^3*(Q_5(zeta)*)^2 mod 5^2. arr6d := [ 0, zeta^3, 7*zeta^3, 18*zeta^3, 24*zeta^3, 2*zeta^3+15+15*zeta, 2*zeta^3+10*zeta^2+20*zeta, 2*zeta^3+20*zeta^2+10, 3*zeta^3+20+20*zeta, 3*zeta^3+5*zeta^2+10*zeta, 3*zeta^3+10*zeta^2+5, 4*zeta^3+10+10*zeta, 4*zeta^3+5*zeta^2+15, 4*zeta^3+15*zeta^2+5*zeta, 6*zeta^3+10+10*zeta, 6*zeta^3+5*zeta^2+15, 6*zeta^3+15*zeta^2+5*zeta, 8*zeta^3+5+5*zeta, 8*zeta^3+15*zeta^2+20, 8*zeta^3+20*zeta^2+15*zeta, 9*zeta^3+20+20*zeta, 9*zeta^3+5*zeta^2+10*zeta, 9*zeta^3+10*zeta^2+5, 11*zeta^3+20+20*zeta, 11*zeta^3+5*zeta^2+10*zeta, 11*zeta^3+10*zeta^2+5, 12*zeta^3+10+10*zeta, 12*zeta^3+5*zeta^2+15, 12*zeta^3+15*zeta^2+5*zeta, 13*zeta^3+15+15*zeta, 13*zeta^3+10*zeta^2+20*zeta, 13*zeta^3+20*zeta^2+10, 14*zeta^3+5+5*zeta, 14*zeta^3+15*zeta^2+20, 14*zeta^3+20*zeta^2+15*zeta, 16*zeta^3+5+5*zeta, 16*zeta^3+15*zeta^2+20, 16*zeta^3+20*zeta^2+15*zeta, 17*zeta^3+20+20*zeta, 17*zeta^3+5*zeta^2+10*zeta, 17*zeta^3+10*zeta^2+5, 19*zeta^3+15+15*zeta, 19*zeta^3+10*zeta^2+20*zeta, 19*zeta^3+20*zeta^2+10, 21*zeta^3+15+15*zeta, 21*zeta^3+10*zeta^2+20*zeta, 21*zeta^3+20*zeta^2+10, 22*zeta^3+5+5*zeta, 22*zeta^3+15*zeta^2+20, 22*zeta^3+20*zeta^2+15*zeta, 23*zeta^3+10+10*zeta, 23*zeta^3+5*zeta^2+15, 23*zeta^3+15*zeta^2+5*zeta, zeta^3+5*zeta+10+10*zeta^2, zeta^3+10*zeta+20+20*zeta^2, zeta^3+15*zeta+5+5*zeta^2, zeta^3+20*zeta+15+15*zeta^2, 2*zeta^3+15*zeta^2+10*zeta+5, 2*zeta^3+5*zeta+5*zeta^2+20, 3*zeta^3+20*zeta^2+5*zeta+15, 3*zeta^3+15*zeta+15*zeta^2+10, 4*zeta^3+10*zeta^2+15*zeta+20, 4*zeta^3+20*zeta+20*zeta^2+5, 5*zeta^3+10*zeta^2+20*zeta+15, 6*zeta^3+10*zeta^2+15*zeta+20, 6*zeta^3+20*zeta+20*zeta^2+5, 7*zeta^3+5*zeta+10+10*zeta^2, 7*zeta^3+10*zeta+20+20*zeta^2, 7*zeta^3+15*zeta+5+5*zeta^2, 7*zeta^3+20*zeta+15+15*zeta^2, 8*zeta^3+5*zeta^2+20*zeta+10, 8*zeta^3+10*zeta+10*zeta^2+15, 9*zeta^3+20*zeta^2+5*zeta+15, 9*zeta^3+15*zeta+15*zeta^2+10, 10*zeta^3+20*zeta^2+15*zeta+5, 11*zeta^3+20*zeta^2+5*zeta+15, 11*zeta^3+15*zeta+15*zeta^2+10, 12*zeta^3+10*zeta^2+15*zeta+20, 12*zeta^3+20*zeta+20*zeta^2+5, 13*zeta^3+15*zeta^2+10*zeta+5, 13*zeta^3+5*zeta+5*zeta^2+20, 14*zeta^3+5*zeta^2+20*zeta+10, 14*zeta^3+10*zeta+10*zeta^2+15, 15*zeta^3+5*zeta^2+10*zeta+20, 16*zeta^3+5*zeta^2+20*zeta+10, 16*zeta^3+10*zeta+10*zeta^2+15, 17*zeta^3+20*zeta^2+5*zeta+15, 17*zeta^3+15*zeta+15*zeta^2+10, 18*zeta^3+5*zeta+10+10*zeta^2, 18*zeta^3+10*zeta+20+20*zeta^2, 18*zeta^3+15*zeta+5+5*zeta^2, 18*zeta^3+20*zeta+15+15*zeta^2, 19*zeta^3+15*zeta^2+10*zeta+5, 19*zeta^3+5*zeta+5*zeta^2+20, 20*zeta^3+15*zeta^2+5*zeta+10, 21*zeta^3+15*zeta^2+10*zeta+5, 21*zeta^3+5*zeta+5*zeta^2+20, 22*zeta^3+5*zeta^2+20*zeta+10, 22*zeta^3+10*zeta+10*zeta^2+15, 23*zeta^3+10*zeta^2+15*zeta+20, 23*zeta^3+20*zeta+20*zeta^2+5, 24*zeta^3+5*zeta+10+10*zeta^2, 24*zeta^3+10*zeta+20+20*zeta^2, 24*zeta^3+15*zeta+5+5*zeta^2, 24*zeta^3+20*zeta+15+15*zeta^2 ]; # The following gives the members of zeta^4*(Q_5(zeta)*)^2 mod 5^2. arr6e := [ 0, 1+zeta5^3+zeta5^2+zeta5, 7+7*zeta5^3+7*zeta5^2+7*zeta5, 18+18*zeta5^3+18*zeta5^2+18*zeta5, 24+24*zeta5^3+24*zeta5^2+24*zeta5, 2*zeta5^2+2*zeta5^3+7*zeta5+17, 2*zeta5^2+22*zeta5+22*zeta5^3+7, 3*zeta5^2+3*zeta5^3+18*zeta5+23, 3*zeta5^2+13*zeta5+13*zeta5^3+18, 4*zeta5^2+4*zeta5^3+24*zeta5+14, 4*zeta5^2+9*zeta5+9*zeta5^3+24, 6*zeta5^2+6*zeta5^3+zeta5+16, 6*zeta5^2+11*zeta5+11*zeta5^3+1, 8*zeta5^2+8*zeta5^3+18*zeta5+13, 8*zeta5^2+23*zeta5+23*zeta5^3+18, 9*zeta5^2+9*zeta5^3+24*zeta5+4, 9*zeta5^2+19*zeta5+19*zeta5^3+24, 11*zeta5^2+11*zeta5^3+zeta5+6, 11*zeta5^2+21*zeta5+21*zeta5^3+1, 12*zeta5^2+12*zeta5^3+7*zeta5+22, 12*zeta5^2+17*zeta5+17*zeta5^3+7, 13*zeta5^2+13*zeta5^3+18*zeta5+3, 13*zeta5^2+8*zeta5+8*zeta5^3+18, 14*zeta5^2+14*zeta5^3+24*zeta5+19, 14*zeta5^2+4*zeta5+4*zeta5^3+24, 16*zeta5^2+16*zeta5^3+zeta5+21, 16*zeta5^2+6*zeta5+6*zeta5^3+1, 17*zeta5^2+17*zeta5^3+7*zeta5+12, 17*zeta5^2+2*zeta5+2*zeta5^3+7, 19*zeta5^2+19*zeta5^3+24*zeta5+9, 19*zeta5^2+14*zeta5+14*zeta5^3+24, 21*zeta5^2+21*zeta5^3+zeta5+11, 21*zeta5^2+16*zeta5+16*zeta5^3+1, 22*zeta5^2+22*zeta5^3+7*zeta5+2, 22*zeta5^2+12*zeta5+12*zeta5^3+7, 23*zeta5^2+23*zeta5^3+18*zeta5+8, 23*zeta5^2+3*zeta5+3*zeta5^3+18, zeta5^3+6*zeta5+11+11*zeta5^2, zeta5^3+11*zeta5+21+21*zeta5^2, zeta5^3+16*zeta5+6+6*zeta5^2, zeta5^3+21*zeta5+16+16*zeta5^2, 2*zeta5^3+7*zeta5^2+22+22*zeta5, 2*zeta5^3+22*zeta5^2+17*zeta5+12, 3*zeta5^3+13*zeta5^2+23*zeta5+8, 3*zeta5^3+18*zeta5^2+13+13*zeta5, 4*zeta5^3+9*zeta5^2+14*zeta5+19, 4*zeta5^3+24*zeta5^2+9+9*zeta5, 5*zeta5^3+15*zeta5^2+10*zeta5+20, 6*zeta5^3+zeta5^2+11+11*zeta5, 6*zeta5^3+11*zeta5^2+16*zeta5+21, 7*zeta5^3+2*zeta5+22+22*zeta5^2, 7*zeta5^3+12*zeta5+17+17*zeta5^2, 7*zeta5^3+17*zeta5+2+2*zeta5^2, 7*zeta5^3+22*zeta5+12+12*zeta5^2, 8*zeta5^3+18*zeta5^2+23+23*zeta5, 8*zeta5^3+23*zeta5^2+13*zeta5+3, 9*zeta5^3+19*zeta5^2+4*zeta5+14, 9*zeta5^3+24*zeta5^2+19+19*zeta5, 10*zeta5^3+5*zeta5^2+20*zeta5+15, 11*zeta5^3+zeta5^2+21+21*zeta5, 11*zeta5^3+21*zeta5^2+6*zeta5+16, 12*zeta5^3+7*zeta5^2+17+17*zeta5, 12*zeta5^3+17*zeta5^2+22*zeta5+2, 13*zeta5^3+8*zeta5^2+3*zeta5+23, 13*zeta5^3+18*zeta5^2+8+8*zeta5, 14*zeta5^3+4*zeta5^2+19*zeta5+9, 14*zeta5^3+24*zeta5^2+4+4*zeta5, 15*zeta5^3+20*zeta5^2+5*zeta5+10, 16*zeta5^3+zeta5^2+6+6*zeta5, 16*zeta5^3+6*zeta5^2+21*zeta5+11, 17*zeta5^3+2*zeta5^2+12*zeta5+22, 17*zeta5^3+7*zeta5^2+2+2*zeta5, 18*zeta5^3+3*zeta5+13+13*zeta5^2, 18*zeta5^3+8*zeta5+23+23*zeta5^2, 18*zeta5^3+13*zeta5+8+8*zeta5^2, 18*zeta5^3+23*zeta5+3+3*zeta5^2, 19*zeta5^3+14*zeta5^2+9*zeta5+4, 19*zeta5^3+24*zeta5^2+14+14*zeta5, 20*zeta5^3+10*zeta5^2+15*zeta5+5, 21*zeta5^3+zeta5^2+16+16*zeta5, 21*zeta5^3+16*zeta5^2+11*zeta5+6, 22*zeta5^3+7*zeta5^2+12+12*zeta5, 22*zeta5^3+12*zeta5^2+2*zeta5+17, 23*zeta5^3+3*zeta5^2+8*zeta5+13, 23*zeta5^3+18*zeta5^2+3+3*zeta5, 24*zeta5^3+4*zeta5+9+9*zeta5^2, 24*zeta5^3+9*zeta5+19+19*zeta5^2, 24*zeta5^3+14*zeta5+4+4*zeta5^2, 24*zeta5^3+19*zeta5+14+14*zeta5^2, 2*zeta5+2*zeta5^2+12+12*zeta5^3, 3*zeta5+3*zeta5^2+8+8*zeta5^3, 4*zeta5+4*zeta5^2+19+19*zeta5^3, 6*zeta5+6*zeta5^2+21+21*zeta5^3, 8*zeta5+8*zeta5^2+3+3*zeta5^3, 9*zeta5+9*zeta5^2+14+14*zeta5^3, 11*zeta5+11*zeta5^2+16+16*zeta5^3, 12*zeta5+12*zeta5^2+2+2*zeta5^3, 13*zeta5+13*zeta5^2+23+23*zeta5^3, 14*zeta5+14*zeta5^2+9+9*zeta5^3, 16*zeta5+16*zeta5^2+11+11*zeta5^3, 17*zeta5+17*zeta5^2+22+22*zeta5^3, 19*zeta5+19*zeta5^2+4+4*zeta5^3, 21*zeta5+21*zeta5^2+6+6*zeta5^3, 22*zeta5+22*zeta5^2+17+17*zeta5^3, 23*zeta5+23*zeta5^2+13+13*zeta5^3 ]; # The following produces an array whose entries are of the # form: [ x-coord, y-coord, first cpt of Cassels map ], # where (x-coord, y-coord) are in Dmon1(Q_5), and the Cassels map q_5 # is being applied to { (x-coord, y-coord), infty } in Jtilde(Q_5). # arr2 := []; for xx from -33 to 33 do if nops([rootp(y^2 - subs(x=xx, Dmon1), 5)]) > 0 then subyy := [msolve( y^2 - subs(x=xx, Dmon1), 5^8 )][1][1]; arr2 := [ op(arr2), [xx,subyy,simplify(subs( x=xx, subyy, rr))] ] fi od; # The following is the image under q_5 of {(1,?), infty}. imagex1 := arr2[14][3] mod 5^2; if imagex1 in arr6a then print(true) fi; if imagex1 in arr6b then print(true) fi; if imagex1 in arr6c then print(true) fi; if imagex1 in arr6d then print(true) fi; if imagex1 in arr6e then print(true) fi; # Note that imagex1 is in none of these, and so it is not in any of: # (Q_5(zeta5)*)^5, zeta*(Q_5(zeta5)*)^5, zeta^2*(Q_5(zeta5)*)^5, # zeta^3*(Q_5(zeta5)*)^5, zeta^4*(Q_5(zeta5)*)^5. # So, now we have rank 0, as expected (2-descent also gave bound 0). # # Here is an overall summary for r=1. # qtilde : Jac(Cmon1)(Q)/ phitilde( Jac(Dmon1)(Q) ) # --> Q(sqrt(5))*/( Q(sqrt(5))* )^5 x Q*/( Q* )^5 # : {(x1,y1),(x2,y2)} |--> # [(y1-sqrt5*(16-12*x1+3*x1^2))*(y2-sqrt5*(16-12*x2+3*x2^2)), # (y1-(16-20*x1+5*x1^2))*(y2-(16-20*x2+5*x2^2))]. # The image of qtilde is a subgroup of: # < [(1-sqrt5)/2, 1], [1, 2], [1, 5] >. # qtilde : T2 = { (0,16), infty } |--> [(1-sqrt5)/2, 1]. # Hence: < [(1-sqrt5)/2, 1] > is a subgroup of the image of qtilde. # # q : Jac(Dmon1)(Q)/ phi( Jac(Cmon1)(Q) ) # --> Q(zeta5)*/( Q(zeta5)* )^5 # x Q(zeta5)*/( Q(zeta5)* )^5 # The image of q is a subgroup of < [zeta, zeta^2] >. # # # Jtilde(Q_2)/phi(J(Q_2)) # J(Q_2)/phitilde(Jtilde(Q_2)) = 5^1. # # Jtilde(Q_2)/phi(J(Q_2)) = 5^1, with image under q_5: < [zeta5,zeta5^2] >. # This was the image of {(-2,y), infty}, with y == 48 (mod 64) under q_5. # # J(Q_2)/phitilde(Jtilde(Q_2)) = 5^0. # Also, j_2 has trivial kernel and # the kernel of jtilde_2 is precisely: < [(1-sqrt5)/2, 1], [1, 5] >. # so that: # j_2^{-1}( q_2( Jtilde(Q_2)/phi(J(Q_2)) ) ) = < [zeta5,zeta5^2] >. # jtilde_2^{-1}( qtilde_2( J(Q_2)/phitilde(Jtilde(Q_2)) ) ) # = [ <(1-sqrt5)/2>, <5> ] = < [(1-sqrt5)/2, 1], [1, 5] >. # # # Jtilde(Q_5)/phi(J(Q_5)) # J(Q_5)/phitilde(Jtilde(Q_5)) = 5^3. # j_5 and jtilde_5 have trivial kernel. # q_5 : {(1,?), infty} |--> elt not even in the image of j_5. # # Jtilde(Q_5)/phi(J(Q_5)) = 5^1. # qtilde_5 : T2 |--> [(1-sqrt5)/2, 1]. # qtilde_5 : {(3,?),infty} |--> [1, 2]. # # Jtilde(Q_5)/phi(J(Q_5)) = 5^2. # so that: # j_5^{-1}( q_5( Jtilde(Q_5)/phi(J(Q_5)) ) ) = < [1, 1] >. # jtilde_5^{-1}( qtilde_5( J(Q_5)/phitilde(Jtilde(Q_5)) ) ) # = < [(1-sqrt5)/2, 1], [1, 2] >. # # Hence: # j_2^{-1}( q_2( Jtilde(Q_2)/phi(J(Q_2)) ) ) # intersection j_5^{-1}( q_5( Jtilde(Q_5)/phi(J(Q_5)) ) ) = < [1, 1] >. # jtilde_2^{-1}( qtilde_2( J(Q_2)/phitilde(Jtilde(Q_2)) ) ) # intersection jtilde_5^{-1}( qtilde_5( J(Q_5)/phitilde(Jtilde(Q_5)) ) ) # = < [(1-sqrt5)/2, 1] >. # Hence we have rank bound zero. # # # My idea for Sha. Let eps1 be fund unit in Q(sqrt(5*p)) # and eps2 be find unit in Q(sqrt(p)). # Choose p = 1 or 4 (mod 5), so that p is a qr mod 5, # forcing T2 = { (0, 16*p^2*sqrt(p)), infty } # to be in J(Q_5), hopefully mapping to [ eps1, 1 ] # (or: [ eps1^2, 1], [ eps^3, 1 ], [ eps^4, 1] ). # Also, choose p so that eps1 is in (Q_p(sqrt(5*p)*)^5 # and in (Q_2(sqrt(5*p)*)^5, which seems easy by HL. # This might find an example, without needing to write a program # for finding the (5,5)-Selmer bound. # Maybe a similar idea could find Sha for the 25-isogeny # on y^2 = x^5 + k. # # The following produces an array: fundunit, whose entries # are of the form [n, fund unit of Q(sqrt(n)]. # It might not be perfect in every case, but we are working # mod 5th powers, so fine as long as the unit is not a 5th power # (can check carefully afterwards if this suggests a promising example). # N := 1000; with(numtheory); fundunit := [ ]; for n from 2 to N do if mobius(n) <> 0 then sofar := false; if n mod 4 = 1 then cfrac( evalf( (1 + sqrt(n))/2, 5000 ), 15, 'con' ); con := [ floor(evalf( (1+sqrt(n))/2 )), op(con) ]; for j from 1 to nops(con) do num1 := numer(con[j]); den1 := denom(con[j]); if sofar = false and abs(num1^2 - n*den1^2) = 1 then fundunit := [ op(fundunit), [ n, num1 + den1*RootOf(t^2-n) ] ]; sofar := true fi; if sofar = false and (num1 mod 2 = 1) and (den1 mod 2 = 1) and abs(num1^2 - n*den1^2) = 4 then fundunit := [ op(fundunit), [ n, num1/2 + den1/2*RootOf(t^2-n) ] ]; sofar := true fi od; cfrac( evalf( sqrt(n), 5000 ), 15, 'con' ); con := [ floor(evalf( sqrt(n) )), op(con) ]; for j from 1 to nops(con) do num1 := numer(con[j]); den1 := denom(con[j]); if sofar = false and abs(num1^2 - n*den1^2) = 1 then fundunit := [ op(fundunit), [ n, num1 + den1*RootOf(t^2-n) ] ]; sofar := true fi; if sofar = false and (num1 mod 2 = 1) and (den1 mod 2 = 1) and abs(num1^2 - n*den1^2) = 4 then fundunit := [ op(fundunit), [ n, num1/2 + den1/2*RootOf(t^2-n) ] ]; sofar := true fi od else cfrac( evalf( sqrt(n), 5000 ), 15, 'con' ); con := [ floor(evalf(sqrt(n))), op(con) ]; for j from 1 to nops(con) do num1 := numer(con[j]); den1 := denom(con[j]); if sofar = false and abs(num1^2 - n*den1^2) = 1 then fundunit := [ op(fundunit), [ n, num1 + den1*RootOf(t^2-n) ] ]; sofar := true fi od fi fi od; # Now recall: # Cmon: y^2 = r*5*(16*r^2 - 12*r*x + 3*x^2)^2 + (x - 4*r)^5 # = r*(16*r^2 - 20*r*x + 5*x^2)^2 + x^5 # Dmon: y^2= 25*r*(-5+2*sqrt5)*(x^2+(15+4*sqrt5)*100*r^2)^2 + (x-10*sqrt5*r)^5 # = 25*r*(-5-2*sqrt5)*(x^2+(15-4*sqrt5)*100*r^2)^2 + (x+10*sqrt5*r)^5 # # T1 = { (4*r, 16*r^2*sqrt(5*r)), infty }. # T2 = { (0, 16*r^2*sqrt(r)), infty }. # qtilde : Jac(Cmon)(Q)/ phitilde( Jac(Dmon)(Q) ) # --> Q(sqrt(5*r))*/( Q(sqrt(5*r))* )^5 x Q(sqrt(r))*/( Q(sqrt(r))* )^5 # : {(x1,y1),(x2,y2)} |--> #[(y1-sqrt(5*r)*(16*r^2-12*x1*r+3*x1^2))*(y2-sqrt(5*r)*(16*r^2-12*x2*r+3*x2^2)), # (y1-sqrt(r)*(16*r^2-20*x1*r+5*x1^2))*(y2-sqrt(r)*(16*r^2-20*x2*r+5*x2^2))]. # Note that anything in the image must have norm a 5th power in Q. This uses: # (y-sqrt(5*r)*(16*r^2-12*x*r+3*x^2))*(y+sqrt(5*r)*(16*r^2-12*x*r+3*x^2)) # = (x - 4*r)^5, # (y-sqrt(r)*(16*r^2-20*x*r+5*x^2))*(y+sqrt(r)*(16*r^2-20*x*r+5*x^2)) # = x^5. # Let r = p == 1 or 4 (mod 5), so that p is a qr mod 5. # Then Q_5(sqrt(5*p)) = Q_5(sqrt(5)) and Q_5(sqrt(p)) = Q_5. # Let w be such that w^2 == p (mod 5). # Then T2 = { (0, 16*r^2*sqrt(r)), infty } = { (0, 16*w^5), infty } # under the embedding Q(sqrt(p)) --> Q_5 : sqrt(p) |--> w, # and T2 = { (0, 16*r^2*sqrt(r)), infty } = { (0, -16*w^5), infty } # under the embedding Q(sqrt(p)) --> Q_5 : sqrt(p) |--> -w, # qtilde_5 : T2 |--> # [ y-sqrt(5*r)*(16*r^2-12*x*r+3*x^2), y+sqrt(r)*(16*r^2-20*x*r+5*x^2) ]. # = [16*r^2*sqrt(r) - 16*r^2*sqrt(r)*sqrt(5), 16*r^2*sqrt(r) + 16*r^2*sqrt(r)] # = [ (1 - sqrt(5))/2, 1 ]. # Take p == 1 or 4 (mod 5). So, our first requirement is that: # (a) (1 - sqrt(5))/2 is the same as eps1^i (fund unit of Q(sqrt(5*p)) # mod 5th powers in Q_5(sqrt(5)), for i = 1,2,3,4. # Also, need # (b) eps1 to be in (Q_p(sqrt(5)*)^5, which seems easy by HL. # For example, suff for eqs1 = a + b*sqrt(5*p) to have a odd, b even. # (c) eps1 to be in (Q_2(sqrt(5)*)^5, which seems easy by HL. # For example, suff for eqs1 = a + b*sqrt(5*p) to have a qr mod p. # # The above might be stated as a lemma. # # The following finds all 5th power powers in Q_5(sqrt(5)) mod 5^3. q5power5 := {}; for i1 from 0 to 5^3-1 do for i2 from 0 to 5^3-1 do expv := simplify( (i1 + i2*sqrt5)^5 ); j2 := coeff(expv, sqrt5); j1 := expv - j2*sqrt5; jj1 := j1 mod 5^3; jj2 := j2 mod 5^3; q5power5 := { op(q5power5), jj1 + jj2*sqrt5 }; od od; # # Let us try r=p=11, the first prime == 1 or 4 (mod 5). # From our array fundunit, the fund unit in Q(sqrt(55)) is: 89 + 12*sqrt(55). # Let this be denoted by: eps1. # Consider Cmon for r=p=11. # Of course, 11 is a qr mod 5 (which is why we chose p == 1 or 4 mod 5): msolve( w^2 = 11, 5^5 ); # The above gave: {w = 3069}, {w = 56}, and we shall use 56. # So: eps1 is just 89 + 12*56*sqrt(5) in Q_5(sqrt(5), modulo an easily # large enough power of 5. simplify( (89 + 12*56*sqrt5)*( (1 - sqrt5)/2 )^3 ) mod 5^3; # The above gave: 68+5*RootOf(_Z^2+120) if 68 + 5*sqrt5 in q5power5 then print(true) fi; # The above gave true, which tells us that it is a 5th mod 5^3, # and that is good enough by HL to be a 5th power in Q_5(sqrt(5)). # (note, if we had taken instead w=3069, then we would have # had simplify( (89 + 12*3069*sqrt5)*( (1 - sqrt5)/2 )^2 ) mod 5^3 a 5th power). # Let's check what 5th power it is: for i1 from 0 to 5^3-1 do for i2 from 0 to 5^3-1 do expv := simplify( (i1 + i2*sqrt5)^5 ); j2 := coeff(expv, sqrt5); j1 := expv - j2*sqrt5; jj1 := j1 mod 5^3; jj2 := j2 mod 5^3; if 68 + 5*sqrt5 = jj1 + jj2*sqrt5 then print(i1 + i2*sqrt5) fi; od od; # The first solution is: 23 + 16*sqrt5. # This tells us that: (23 + 16*sqrt5)^5 == eps1*( (1 - sqrt5)/2 )^3 mod 5^3, simplify( (23 + 16*sqrt5)^5 - (89 + 12*56*sqrt5)*( (1 - sqrt5)/2 )^3 ) mod 5^3; # It must then be that also eps1^2*( (1 - sqrt5)/2 ) is a 5th power. # Check directly: simplify( (89 + 12*56*sqrt5)^2*( (1 - sqrt5)/2 ) ) mod 5^3; # The above gave: 68+75*RootOf(_Z^2+120) if 68 + 75*sqrt5 in q5power5 then print(true) fi; # The above gave true. # Let's check what 5th power it is: for i1 from 0 to 5^3-1 do for i2 from 0 to 5^3-1 do expv := simplify( (i1 + i2*sqrt5)^5 ); j2 := coeff(expv, sqrt5); j1 := expv - j2*sqrt5; jj1 := j1 mod 5^3; jj2 := j2 mod 5^3; if 68 + 75*sqrt5 = jj1 + jj2*sqrt5 then print(i1 + i2*sqrt5) fi; od od; # The first soln is: 18 + 15*sqrt5: simplify( (18 + 15*sqrt5)^5 - (89 + 12*56*sqrt5)^2*( (1 - sqrt5)/2 ) ) mod 5^3; # so that f(x) = x^5 - eps1^2*( (1 - sqrt5)/2 ) has: # | f( 18 + 15*sqrt5 ) |_5 <= 5^{-3}, and: # | f'( 18 + 15*sqrt5 ) |_5 = 5^{-1}, # so by HL, f(x) has a soln in Q_5, so that eps1^2*( (1 - sqrt5)/2 ) # is in (Q_5(sqrt5)*)^5. It follows that (1 - sqrt5)/2 = eps1^3. # in Q_5(sqrt5)*/(Q_5(sqrt5)*)^5. # So, let T2 = { (0, 16*11^2*sqrt(11)), infty } be chosen # to be: T2_5 = { (0, 16*w^5), infty } in J(Q_5), where w is in Q_5, # with w^2 = 11 and w == 1 (mod 5). Then: # qtilde_5 : T2_5 |--> [ eps1^3, 1], # and so 2*T2_5 |--> [ (eps1^3)^2, 1 ] = [eps1, 1] (mod 5th powers). # So far, this tells us that [ eps1, 1] cannot be disqualified # by 5-adic arguments. # If we want to be carefully literal about qtilde_5 then strictly # speaking qtilde_5 : T2_5 |--> [(w^5 - w^4*sqrt(55))/2, 1] # where we are taking approx w = 56 in Q_5 (w^2 = 11 and w == 1 (mod 5). # Then (adjusting our previous Maple command by absorbing 56*sqrt5 # as sqrt55): sqrt55 := RootOf(z^2 - 55); simplify( (56*18 + 15*sqrt55)^5 - (89 + 12*sqrt55)^2*( (56^5 - 56^4*sqrt55)/2 ) ) mod 5^3; # The above shows directly that T2_5 maps to eps1^3. # # # Now, let f(x) = x^5 - eps1 = x^5 - (89 + 12*sqrt(55)) over Q_2(sqrt55). # Then | f(1) |_2 < 1 and | f'(1) |_2 = 1, so soln by HL. # This means that [ eps1, 1 ] is in the kernel of jtilde_2 # and cannot be disqualified by 2-adic arguments. # # Now, let f(x) = x^5 - eps1 = x^5 - (89 + 12*sqrt(55)) over Q_11(sqrt55). # Then | f(1) |_11 < 1 and | f'(1) |_11 = 1, so soln by HL. # This means that [ eps1, 1 ] is in the kernel of jtilde_11 # and cannot be disqualified by 11-adic arguments. # # Clearly nothing can be disqualified at infinity, since R*/(R*)^5 # is the trivial group and everything is in the kernel of jtilde_infty. # # This means the [ eps1, 1 ] is in the Selmer group of the (5,5) # isogeny, and # J(Q)/phitilde(Jtilde(Q)) is bound above by 5^1. # On the other hand, J[tilde](Q) and indeed all of J[5](Q) is # the trivial group here, so that the (5,5)-Selmer bound # on the rank is nonzero. # # The above should be easy to automate, if I wanted to get # other examples where the (5,5)-Selmer bound is nonzero. # Note also that the y^2 = x^5 + k family might be amenable to similar args. # # The following in Magma give 2-Selmer bound 0, so it appears # that we have 5-part of Sha in this case. # Most likely this can be generalised to a congruence class of p, # with conditions # _ := PolynomialRing(Rationals()); # r := 11; # Cmon := 256*r^5-640*x*r^4+560*x^2*r^3-200*x^3*r^2+25*x^4*r+x^5; # rkCmon := TwoSelmerGroupData( Jacobian( HyperellipticCurve( Cmon ) ) ); # rkCmon; # TorsionSubgroup( Jacobian( HyperellipticCurve( Cmon ) ) ); # The above gave: 2-Selmer bound of 0 on the rank, and trivial torsion group. # Note: this means there cannot be any affine points in J(Q). # I might check whether there are affine points everywhere locally.