Bill Allombert on Wed, 30 May 2012 13:26:27 +0200

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Re: ellgenerators

On Wed, May 30, 2012 at 02:35:03PM +0400, Max Alekseyev wrote:
> Why pari cannot compute generators of an elliptic curve, while sage can?
> I thought they have roughly the same "power" w.r.t. elliptic curves.
> ? ellgenerators(ellinit([-35750837071872,63237970697887678464]))
>   ***   at top-level: ellgenerators(ellini
>   ***                 ^--------------------
>   *** ellgenerators: error opening elldata file:
> `/usr/local/share/pari/elldata/ell1816'.
>   ***   Break loop: type 'break' to go back to GP
> sage: EllipticCurve([-35750837071872,63237970697887678464]).gens()
> [(-2365632 : 11600547840 : 1)]

I suppose Sage is using John Cremona code (mwrank) which is not available in PARI.
We are slowly implementing the required algorithms. PARI instead read the generators
from the database.

In the development version, there is a function ellheegner which is able to find
a non-torsion point on a curve of analytic rank 1 and small conductor.

In you example, PARI found a point
? ellheegner(ellinit([-35750837071872,63237970697887678464]))
%3 = [206471808,2965585996800]
  ***   last result computed in 17,193 ms.

which is a generator in this case.

You can alos use Denis Simon 2-descent script which find yet another generator:
[7473600,14611166208] (there are 8 equivalent generators) by a different method.

Obviously, we would gladly accept any help with this project.