Igor Schein on Wed, 3 May 2000 15:45:25 -0400

 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.