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

[Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>]

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