Re: trying to use qflllgram on the height matrix of elliptic curve points sometimes gives odd results

[American Citizen <website.reader3@gmail.com>]

Bill:

Thank you for the comments. I will need to change my gp-pari code for 
elliptic curves and reducing the points to a Mordell-Weil basis or subset .

Appreciate your explanation of qflllgram() and how it works which 
explains why my code hung up.

Randall

On 2/10/24 13:47, Bill Allombert wrote:
> On Sat, Feb 10, 2024 at 12:50:04PM -0800, American Citizen wrote:
> > Hello everyone:
> >
> > I have a certain rank=4 Z2xZ6 curve, from an isogenous Z12 rank 4 curve
> > discovered by Tom Fisher in 2008, which I am currently using to analyze
> > certain points on the curve, but these points are compositions of the
> > Mordell-Weil basis points which are known.
> >
> > The basic idea is this
> >
> > h = ellheightmatrix(E,[p[1], p[2], p[3], p[4], q]);
> >
> > mattranspose( qflllgram(h) )
> >
> > But sometimes the matrix does NOT come out right, and occasionally the gp
> > pari program hangs up and then starts to add to the heap requesting
> > gigabytes of ram stuck on apparently trying to do the qflllgram factoring.
> Yes. As a rule one should be very careful when using qflllgram on inexact
> matrices, because while mathematic garantee the height is a positive defined
> quadratic form, it does not garantee that its Gram matrix approximated to
> some accuracy is defined positive.
>
> Running LLL on  non-defined positive matrices might go in a infinite
> loop, because it finds larger and larger vectors whose norm is converging to 0.
>
> > Questions:
> >
> > Is qflllgram(ellheightmatrix(e,pts)) the right way to discover the points
> > composition of the unknown point q and the MW basis p1-4 ?
> Probably not. Why not just use matsolve ?
> If the result is not integral or incorrect, increase the precision.
>
> M=ellheightmatrix(E,[p[1], p[2], p[3], p[4]]);
> V=vector(4,i,ellheight(E,p[i],q));
> matsolve(M,V)
>
> Cheers,
> Bill