Re: ellintpoints(E) workaround

[hermann@stamm-wilbrandt.de]

On 2026-10-07 10:49, Bill Allombert wrote:
> On Wed, Oct 07, 2026 at 12:52:18AM +0200, hermann@stamm-wilbrandt.de 
> wrote:
> > sage(cmd) = eval(externstr(Str("sage -q -c \"", cmd, "\""))[1]);
> >
> > ellintpoints(E) = {
> >   sage(Str("print([[int(P[0]), int(P[1])] for P in EllipticCurve(",
> >        E[1..5],").integral_points()])"));
> > }
>
> How does it work ? How far can it go ?
I do not know yet.

> For conductor at most 500000, one can use John Cremona database:
> https://github.com/JohnCremona/ecdata/tree/master/intpts
The curve's conductor is too high (if I determined it correctly):

? E=ellinit([-14776, -670031]);
? ellglobalred(E)[1]
447294352204
?

> If one do not need a proven result one can do ellratpoints(E,[10^9,1])
>
> ? E=ellinit([-14776, -670031]);
> ? ellratpoints(E,[10^9,1])
> %8 = 
> [[-80,7],[-80,-7],[-73,140],[-73,-140],[-66,133],[-66,-133],[-60,23],[-60,-23],[140,73],[140,-73],[214,2443],[214,-2443],[227,2770],[227,-2770],[246,3253],[246,-3253],[514,11293],[514,-11293],[594,14147],[594,-14147],[802,22435],[802,-22435],[1660,67447],[1660,-67447],[31224,5517337],[31224,-5517337],[100016,31630343],[100016,-31630343],[160360,64216103],[160360,-64216103]]
>
> Cheers,
> Bill
Thanks, that works.

Because I searched for integral points I would not have looked at 
ellratpoints.
Doc says to look for hyperellratpoints for description of h.
Seeing
"- a vector [N, D] with D ≤ N : find the points [n/d, y] with |n| ≤ N , 
d ≤ D."
I know understand that 2nd vector entry 1 means only integral points.

But sagemath E.integral_points() is called without height.
It seems to use Baker bounds, not sure.
But how to choose H for ellratpoints(E,[H,1])?


I needed to determine the generators, but ellgenerators(E) failed.
Even after I had installed pari-elldata package.
Is the workaround I found guaranteed to work?
https://gist.github.com/Hermann-SW/9925e91bd68a5d2c14f95c9940448eab#file-integral_points_coefficients-gp-L20-L21

Regards,

Hermann.