Re: general question relating to elliptic curves and their rational points and creating a pool of rational points of count N

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

Bill

Thanks for that function for b, that is exactly what I needed. I 
appreciate the post also because of the mention of lattice theory, 
interesting!

Randall

On 4/15/24 03:00, Bill Allombert wrote:
> On Sun, Apr 14, 2024 at 05:51:24PM -0700, American Citizen wrote:
> > Hello:
> >
> > I am currently working with rational Diophantine sextuples, example: [5/4,
> > 5/36, 32/9, 189/4, 665/1521, 3213/676] where the product of any pair + 1 is
> > a rational square.
> >
> > If we select 3 of the ratios, making a triple, say [5/4, 5/36, 32/9], we can
> > create an elliptic curve associated with the triple [a,b,c]
> >
> > E:  [0,(a*b+a*c+b*c),0,(a*b*c)*(a+b+c),(a*b*c)^2]
> >
> > I have been working with 758 sextuples, and found elliptic curves of rank =1
> > through rank = 10 for Diophantine triples derived from the sextuples. (20
> > triples per sextuple).
> >
> > My question to the group is this, given the Mordell-Weil basis (which seems
> > quite easy to find for E), how can I determine a cut-off value of elliptic
> > curve point heights for creating a pool of 10K points that are on the curve?
> >
> > I am using a gp-pari command called "ellpool(E,p,h)" where E is the curve, p
> >
> > I need to determine h such that the count of the pool is around 10K or so.
> > Determining h seems somewhat hard to figure out, although I was using the
> > regulator^(1/rank) * F to set the height h (where F is around 5 to 12 or so)
> > and that generally works, but not always. Of  course the torsion group for
> > the curve E plays a huge part too in finding the pool of points, as the
> > torsion points automatically increase the rational points found.
> >
> > Has someone done this work before? Can any papers be cited?
> This is just come from lattice theory:
>
> M=ellheightmatrix(E,MW_basis)
> gives you a lattice and you can use qfminim(M,B,,2) to find short vectors in
> your lattices.
>
> Each lattice vectors will give you #tors points having the same height.
> So pick
> N=10000/elltors(E)[1]
>
> What you need to do then is to pick B so that qfminim return about N vectors.
>
> The expected relation between N and B is something like this:
>
> B = sqrtn((N /(Pi^(n/2) / gamma(n/2 + 1)))^2*reg,n)
> when n is the rank.
>
> E=ellinit([0, 6625/1296, 0, 2225/729, 2500/6561])
> MW_basis = [[124/9, 4879/81], [1600/9, 194750/81]]
> M=ellheightmatrix(E,MW_basis)
> n=#MW_basis; reg = matdet(M);
> tors=elltors(E);
> N=10000/tors[1];
> B = sqrtn((N /(Pi^(n/2) / gamma(n/2 + 1)))^2*reg,n)
> qfminim(M,B,,2)
>
> %8 = [2500,2826.9854922754748719832305224795906320
>
> We are very lucky, we get exactly the number we asked for!
>
> Cheers,
> Bill.