This currently contains only those ranks for which
descriptions of the computations have been published.
I shall at some stage include other ranks which have
been computed, once I have decided on a good reader-verification
format. That is to say, a concise piece of information
which the reader can use to check the correctness
of the rank computation.
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The following are from the paper:
Gordon, D.M. \& Grant, D. (1993) Computing the Mordell--Weil rank of
jacobians of curves of genus~2. {\it Trans.\ Amer.\ Math.\ Soc.}
{\bf 337}, 807--824.
(1). Y^2 = X*(X-1)*(X-2)*(X-5)*(X-6)
Jacobian's rational torsion is the 2-torsion group of size 16.
Jacobian has rank 1, with infinite-order divisor {(3,6), infty}.
(2). Y^2 = X*(X-3)*(X-4)*(X-6)*(X-7)
Jacobian's rational torsion is the 2-torsion group of size 16.
Jacobian has rank 0.
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The following ten curves are from the paper:
Flynn, E.V. (1994) Descent via isogeny in dimension 2.
{\it Acta~Arith.} {\bf 66}, 23--43.
(3). Y^2 = (X^2+6*X+7)*(X^2+4*X+1)*(X^2+2*X+3)
Jacobian's rational torsion is the 2-torsion group of size 4.
Jacobian has rank 2, with independent infinite-order divisor:
{(-2,3),(-2,3)} and
{(-1+sqrt(6),16+8*sqrt(6),-1-sqrt(6),16-8*sqrt(6))}
A curve with isogenous jacobian is:
Y^2 = (X^2-2*X-5)*(X^2+2*X-1)*(X^2+6*X+11)
Jacobian's rational torsion is the 2-torsion group of size 4.
Jacobian has rank 2, with independent infinite-order divisor:
{(-5/2,15/8),(-5/2,15/8)} and
{(-5/3+sqrt(-2)/3,32/27+80*sqrt(-2)/27),(-5/3-sqrt(-2)/3,32/27-80*sqrt(-2)/27)}
(4). Y^2 = k*(X^2+1)*(X^2+2)*(X^2+X+1)
and curve with isogenous jacobian:
Y^2 = k*(X^2-2*X-2)*(X^2-1)*(-2*X)
[N.B. In the article there was a typo: the final "2" in the
second curve was missing].
When k=1,2,6, the jacobian of each curve has rank 1.
When k=-2, the jacobian of each curve has rank 0.
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The following example is from:
Flynn, E.V., Poonen, B. \& Schaefer, E.F. (1995)
Cycles of quadratic polynomials and rational points on a genus~2
curve. Preprint.
(5). Y^2 = X^6 + 8*X^5 + 22*X^4 + 22*X^3 + 5*X^2 + 6*X + 1
Jacobian has no rational torsion.
Jacobian has rank 1, with infinite divisor: {infty+, infty+}
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