From: tom@womack.net (Tom Womack)
Subject: Curves with interesting torsion groups
Date: 13 Jul 00 13:14:06 GMT
Newsgroups: sci.math.numberthy
Summary: [missing]

Ever wanted to know the parameterisation for elliptic curves over Q with
Z12-torsion?

Or an example of a curve with 7-torsion and Sha=25?
(y^2 + xy = x^3 - 1059708046x - 147410635402108 works)

Well, I haven't either, but using a table in a paper of Kubert in Proc:
London Math Society 1976 p.217 and some analytic-rank code I wrote in Pari,
I've done some searches over curves with interesting torsion groups, and
produced tables of smallest conductor at given rank and torsion, and
smallest conductor for a rank-0 curve of given torsion and sha.

http://tom.womack.net/maths/torsion.htm is the place to go, and links to
torseg.gp which is a Pari implementation of the parameterisations for all 14
possible non-trivial torsion groups; the tables only give the 11 torsion
groups which are one-parameter families.

Tom