For a general representation $\rho: G_{\mathbb{Q}} \rightarrow \operatorname{GL}(V)$, where $V$ is a two dimensional $\overline{\mathbb{F}}_p$ vector space, the level $N(\rho)$ in Serre's conjecture is the product over all $\ell \neq p$ of $\ell^{n(\ell,p)}$ where $n(\ell,p)$ is given by the usual formula. I want to apply this to elliptic curves, and prove that if $E$ is a semistable elliptic curve with discriminant $\Delta$ then $$ N(\rho) = \prod_{\substack{ \ell \neq p \\ p \nmid \operatorname{ord}_\ell(\Delta) }} \ell $$ Every source I have consulted shows that $\rho$ is unramified at $\ell$ (when $E$ has good reduction at $\ell$), and a similar (though harder) argument whe $E$ has multiplicative reduction at $\ell$.
1) However, it seems to me that if $\rho$ is unramified, then it just means that $n(\ell,p) = 0$ as the inertia group (which is the ramification group $G_0$) is trivial. So the proof does not account for the cases where $n(\ell,p) = 1$ and I am not sure where these come from.
2) Does anyone know of concrete examples (or references) where we can calculate $N(\rho)$, $k(\rho)$ and the cuspidal eigenform $f$ of level $N(\rho)$, weight $k(\rho)$ directly, without using a computer (for example to compare the $q$-coefficients of the eigenform thus obtained to the traces of Frobenius elements $a_\ell$ on the $p$-torsion of $E$).
Thanks!
If $E$ is semistable at $\ell$, then $n(l,p) = 0$ or $1$, depending on whether or not $p$ divides the number of connected components of the Neron model of $E$ mod $\ell$. (This is expressed concretely in terms of $ord_{\ell}(\Delta)$.)
The unramified case follows from the criterion of Neron--Ogg--Shafarevic.
The ramified case is most easily proved by using the description of the semistable curve $E$ as a Tate curve over $\mathbb Q_{\ell}$. (Then $ord_{\ell}(\Delta)$ will appear in the guise of $ord_{\ell}(q)$, where $q$ is the period of the Tate curve.)
As for examples, what do you have in mind exactly?
Firstly, have you looked at tables of elliptic curves? These typically will give you an equation for the curve, so that (if you wanted) you could compute the $a_{\ell}$'s by counting points mod $\ell$.
Then you could compare with tables of modular forms.