Bill Allombert on Wed, 15 May 2013 11:43:43 +0200

 Re: ellap problem

On Wed, May 15, 2013 at 05:08:01PM +0800, choy valerie wrote:
> Hi,
>
> I was trying to compute the cardinality of a 512-bit elliptic curve. For example, for y^2 = x^3 + a4.x + a6 mod p where
>
> p=10566623376041669505825220895462627801380145726624712771836144280024219722297939525451022774579043146020265329009462778097121538072213487555318041328039599
>
> a4=6557325753041215216697541661243177462316789827714644669153173743634615853451990973713919498189870050066222585898470391225455349541996119434962754618249308
> a6=3887529832007272230349363633177495741990999092935596186265775681498612008751646149414901742385996951003068331698140100902826619262789957720578797201032523
>
> When I tried to compute the order directly using GP with:
>
> sage: E = gp.ellinit([0,0,0,a4,a6])
> sage: ap = E.ellap(p)
>
> I encountered the error:
>
>
> *** at top-level: sage[9] = ellap(sage[7],sage[8]
> ***                                   ^----------------------
> *** ellap: not enough modular polynomials

The result is
126651527956688227530515631165506157823294213416153290335354701023424851011267

> May I know how to resolve this problem? Any help is greatly appreciated!

This is not a PARI/GP problem but a sage problem. To perform this computation you
need the modular polynomials upto l = 229.
They are provided in the extra PARI package seadata. If sage do not provide them,
PARI cannot do the computation.
Maybe there is an optionnal SAGE package you need to install to get the full