Karim BELABAS on Thu, 4 May 2000 13:25:47 +0200 (MET DST)

 Re: large rank and torsion group

```[Igor:]
> 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
>
> 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.
>

Which GP version, architecture ? And finally, on which precise input ? Given
the functions

E(a,b,c) = [0, (b + c)*a + c*b, 0, c*b*a^2 + (c*b^2 + c^2*b)*a, c^2*b^2*a^2]
[rationaly equivalent to the original one]

Tors(a,b,c) = elltors(ellinit(E(a,b,c)), 1)
test() = Tors(119/60, 3398759/864000, -864000/3398759)

In current CVS + 32bit UltraSparc, I get the following results :

(13:08) gp > \p28
realprecision = 28 significant digits
(13:08) gp > test
***   precision too low in torselldoud.

after increasing the precision, I get the correct result:

(13:08) gp > \p100
realprecision = 105 significant digits (100 digits displayed)
(13:08) gp > test
/         / / 3758759 11354445338831 \ / 14400   \ \ \
%1 = | 8 (4 2) | | ------- -------------- | | ----- 0 | | |
\         \ \ 1216800 1480602240000  / \ 28561   / / /

(with prettyprinter off, this reads: [8, [4, 2], [[3758759/1216800,
11354445338831/1480602240000], [14400/28561, 0]]] )

On my machine, all three examples react in the same way, yielding the
expected result at \p100.

Cheers,

Karim.

P.S: The old Nagell-Lutz implementation is unable to give the result in
decent time (stopped after one hour, needed 160MB of stack).
__
Karim Belabas                    email: Karim.Belabas@math.u-psud.fr
Dep. de Mathematiques, Bat. 425
Universite Paris-Sud             Tel: (00 33) 1 69 15 57 48
F-91405 Orsay (France)           Fax: (00 33) 1 69 15 60 19
--