From: duje@math.hr (Andrej Dujella)
Subject: rank=8, torsion group=Z/2Z*Z/2Z
Date: 26 Apr 99 14:52:55 GMT
Newsgroups: sci.math.numberthy
Keywords: rationals such that the product of any two is one less than a square

I found three examples of elliptic cuves over Q
with torsion group isomorphic to Z/2Z * Z/2Z
and with rank = 8.
This improves my previous examples with rank = 7
(see A. Dujella: Diophantine triples and construction of
high-rank elliptic curves over Q with three non-trivial
2-torsion points, Rocky Mountain J. Math., to appear).

These elliptic curves are

           y^2=x*[x+(b-a)(d-c)]*[x+(c-a)(d-b)],

where (a,b,c,d)=
(32/91, 60/91, 1878240/1324801, 15343900/12215287),
(17/448, 2145/448, 23460/7, 2352/7921) and
(559/1380, 252/115, 24264935/2979076, 16454108/1703535).

The quadruples {a,b,c,d} are subsets of rational Diophantine sextuples
(sets of six positive rationals such that the product of any two
of them is one less than a square) discovered by Philip Gibbs
(P. Gibbs: Some rational Diophantine sextuples, math.NT/9902081;
P. Gibbs: A generalised Stern-Brocot tree from regular Diophantine
quadruples, math.NT/9903035).

The ranks are computed with John Cremona's program MWRANK.

        Andrej  Dujella

        Department of Mathematics
        University of Zagreb
        Bijenicka cesta 30
        10000 Zagreb
        CROATIA

        http://www.math.hr/~duje/
==============================================================================
[The question of  Diophantine sextuples  has arisen before. See
	97/all_xy_are_z2-1    --djr]