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 $p$ 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 $\mathbb{Z}$).

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 $\mathbb{Z}$).