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

[Igor Schein <igor@txc.com>]

On Fri, Jun 04, 1999 at 04:10:38PM -0400, Igor Schein wrote:
> 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

On the other hand, if I use the integral model
and reduce the curves globally, it works fine:

? 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];
? d=[15343900/12215287,2352/7921,16454108/1703535];
? cubic(n)=f(a[n],b[n],c[n],d[n]);
? Ell(n)=local(t);t=cubic(n);[0,polcoeff(t,2),0,polcoeff(t,1),polcoeff(t,0)];
? changecurve(v)=local(E);E=ellinit(v);E=ellchangecurve(E,ellglobalred(E)[2]);[E[1],E[2],E[3],E[4],E[5]];
? elltors(ellinit(changecurve(Ell(1))))
[4, [2, 2], [[293424555922235/4, -293424555922235/8], [58479762744360, -29239881372180]]]
? elltors(ellinit(changecurve(Ell(2))))
[4, [2, 2], [[7516680352875/4, -7516680352879/8], [2223647732595, -1111823866298]]]
? elltors(ellinit(changecurve(Ell(3))))
[4, [2, 2], [[2813046327973891583/4, -2813046327973891583/8], [812142518073808192, -406071259036904096]]]

So the bug is definitely related to handling non-integral
coefficients.

Thanks

Igor