TITLE: Neron models and ramification
QUESTION [9 upvotes]: I encountered this result while reading a few things, and it was stated without reference. I am having a hard time finding a reference for it (or a simple proof), so maybe you can help me:
let $E$ be an elliptic curve over $\mathbb{Q}$, $\ell$ a prime number, and consider $\rho=\rho_{E,\ell}$ the Galois representation of $Gal(\bar{\mathbb{Q}}/\mathbb{Q})$ on the $\ell$-adic Tate module of $E$. Let $p\neq\ell$ be a finite prime number where $\rho$ is ramified. Let $\ell^n$ be the highest power of $\ell$ such that $\rho\mod\ell^n$ is unramified at $p$. 
Then $(\ell)^n=(c_p)$ in $\mathbb{Z}_{\ell}$, where $c_p$ is the Tamagawa factor of $E$ at $p$ (the order of the component group of the special fiber at $p$ of the N\'eron model of $E$ over $\mathbb{Z}$).

REPLY [3 votes]: If $E$ has (split) multiplicative reduction, then you can probably get this by a direct computation on the Tate model. And for the additive reduction cases, one could possibly use explicit models for the various reduction types, a la Tate's algortithm, and do a direct computation. I expect this is fairly easy if $p\ge5$, reasonable if $p=3$, and quite painful (if even possible) for $p=2$. So the SGA reference that Olivier gives is undoubtedly the right approach, but if you want to avoid all the machinery, this might give a way to do it.