Re: defining elliptic curves over a finite field

[Bill Allombert <Bill.Allombert@math.u-bordeaux1.fr>]

On Wed, Nov 09, 2011 at 04:51:50PM +0000, McLaughlin, James (MATH) wrote:
> How do I get pari to define an elliptic curve over a finite field F_p (say F_5)?
> The instructions for ellinit have nothing to say about this. The manual several talks about curves being defined over a prime field F_p, but does not actually say how to do it.
> 
> Example: y^2=x^3+x+1
> I do            ec=ellinit([0,0,0,1,1])
> If I do  ellad(ec,[0,1],[01])
> the output is [1/4,-9/8] (so the arithmetic appears to be over Q).
> 
> How can I define the curve so the output treats the curve as being over F_5 and outputs [4,2]?

You need to give coefficients in F_5.

Try
ec=ellinit([0,0,0,1,1]*Mod(1,5))

Cheers,
Bill.