Igor Schein on Wed, 3 May 2000 15:45:25 -0400 |
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Re: large rank and torsion group |
On Wed, May 03, 2000 at 03:07:30PM -0400, Andrej Dujella wrote: > I am searching for Diophantine triples (i.e. rationals a,b,c > such that ab+1, ac+1 and bc+1 are all perfect squares) > with the property that the corresponding elliptic curve > > y^2=(ax+1)(bx+1)(cx+1) > > has large rank and/or large torsion group. > > I found the following (interesting) examples: > > {a,b,c}={217/69, 355/69, 368851912/328509} > torsion group = Z/2Z * Z/2Z, rank = 8 > > {a,b,c}={119/60, 3398759/864000, -864000/3398759} > torsion group = Z/2Z * Z/4Z, rank = 5 > > {a,b,c}={145/408, -408/145, -145439/59160} > torsion group = Z/2Z * Z/8Z, rank = 3 > > I would like to know what are current records for the ranks of > elliptic curves over Q with torsion groups Z/2Z * Z/mZ for m=2,4,8. > > > Andrej Dujella > duje@math.hr Hi, I couldn't resist checking the above with gp, and guess what, gp shows torsion group = Z/2Z * Z/2Z, for the 2nd curve, which is not what Andrej claims. Remembering that there was a problem with the old implementation of elltors(), I tend to suspect a precision problem here. Any comments? Thanks Igor