Re: Elliptic curves over finite fields

[Karim Belabas <Karim.Belabas@math.u-psud.fr>]

* Mc Laughlin, James [2004-06-02 21:33]:
> I use to have access to magma and there was a command there that would output what I wanted directly- something like the following, for example:
> p:=1123;
> r:=3;
> E := EllipticCurve([GF(p,r) | 0, 0, 0, 0, 19 ]);
> ord:=Order(E);
>
> ord would be what I wanted.
[...]
> I know a few extra lines of code would do what I wanted. What I wondered
> was if there a command like "Order" in Magma which would do the same thing.
> Seems like there should be.

Currently, ellap only works for curves defined over the prime field F_p. 
Introducing a more general function [ or complicating the interface ] is
not really useful if it only treats the simplest case, covered by a GP
one-liner. I agree general finite fields should be supported, but in a
systematic way.

None of the (few) current developers are currently working on the 'elliptic
curve' module, but I'm more than willing to accept general implementations
of point-counting algorithms over finite fields ( not necessarily
record-breaking ones :-)

So far, I'm aware of an optimized GP implementation of SEA (over F_p),
which has not been submitted yet, and a straightforward implementation of
Satoh, which can be accessed from my Contributed scripts page

  http://www.math.u-psud.fr/~belabas/pari/scripts/

Cheers,

    Karim.
--
Karim Belabas                     Tel: (+33) (0)1 69 15 57 48
Dep. de Mathematiques, Bat. 425   Fax: (+33) (0)1 69 15 60 19
Universite Paris-Sud              http://www.math.u-psud.fr/~belabas/
F-91405 Orsay (France)            http://pari.math.u-bordeaux.fr/  [PARI/GP]