Igor Schein on Fri, 4 Jun 1999 16:10:38 -0400

 Re: rank=8, torsion group=Z/2Z*Z/2Z

```On Mon, Apr 26, 1999 at 10:52:55AM -0400, Andrej Dujella wrote:
> 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).
[snip]

I decided to verify this with development version of PARI, with
Doud's algorithm implemented:

? f(a,b,c,d)=x*(x+(b-a)*(d-c))*(x+(c-a)*(d-b));
? a=[32/91,17/448,559/1380];
? b=[60/91,2145/448,252/115];
? c=[1878240/1324801,23460/7,24264935/2979076]
[1878240/1324801, 23460/7, 24264935/2979076]
? d=[15343900/12215287,2352/7921,16454108/1703535]
[15343900/12215287, 2352/7921, 16454108/1703535]
? cubic(n)=f(a[n],b[n],c[n],d[n]);
? Ell(n)=local(t);t=cubic(n);ellinit([0,polcoeff(t,2),0,polcoeff(t,1),polcoeff(t,0)]);
? elltors(Ell(1))
[2, [2], [[0, 0]]]
? elltors(Ell(2))
[2, [2], [[0, 0]]]
? elltors(Ell(3))
[2, [2], [[0, 0]]]

So PARI thinks the torsion groups are isomorphic to Z/2Z,
contrary to what Dr. Dujella's findings.

I looked at it many times, and I believe I set it up correctly.  So
looks like it's a PARI bug.

Thanks

Igor
```