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Let $E_1$ and $E_2$ be elliptic curves over $\mathbb{Q}$ with good, ordinary reduction at an odd prime $p$. I'm wondering how to determine whether $E_1[p]$ and $E_2[p]$ are isomorphic $\mathbb{F}_p[G_\mathbb{Q}]$-modules.

It is my understanding that I can check this by looking for a congruence mod $p$ between the associated modular forms, but I'm not exactly sure what constitutes a congruence. I guess that it should mean that almost all of the numbers $a_\ell$ for the two curves (for $\ell\neq p$ a prime of good reduction for both $E_1$ and $E_2$) are congruent mod $p$. But this confuses me for the following reason. These $a_\ell$ are the traces of Frobenii on the $p$-adic Tate module, and I assume the reason this idea should work is that if they are (almost) all congruent mod $p$, then the $\mathbb{F}_p$-representations should be isomorphic by considerations with Chebotarev density (assuming these representations are semisimple, a fact which I think\hope is true but for which I have no reference).

If this is indeed what is meant by a congruence, then how could I check it in practice? I can look up my curves in Cremona's tables and look at as many of coefficients of the $q$-expansions of the modular forms as I want, but how many do I have to look at before I conclude that the congruence holds?

Disclaimer: I am very much new to computational stuff, so if I have said something naive or borderline ridiculous, I apologize. I'm more accustomed to working sort of...theoretically.

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I believe what you are looking for is called "the Sturm bound". See, for example, William Stein's "Modular forms, a computational approach" (which is freely available online). In particular, read Section 9.4, on congruences between newforms. Sturm's theorem is Theorem 9.18 in that section, and the applications to congruences appear in Corollaries 9.19 and 9.20, and Theorems 9.21 and 9.22.

If you are new to computing with modular forms, Stein's book is a very nice reference on the subject.

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    I guess actually the only prime$I$can see where the coefficients aren't congruent is $7$, which is also the only prime dividing the conductor of $E_2$ but not the conductor of $E_1$. I didn't think about the fact that you have to put them both in the same space of cusp forms before comparing them...I guess this is my problem.2012-04-05