I'd like a reference (e.g. something published somewhere that I can cite in a paper) for the proof of the following:

Let $E$ be an elliptic curve over $\mathbb Q$ with minimal discriminant $\Delta$, let $p$ be a prime, at which $E$ has potentially multiplicative reduction and let $\ell$ be a prime different than $p$. Then the mod $\ell$ representation is unramified at $p$ iff $\ell$ divides the valuation of $\Delta$ at $p$.

This is used for instance in the proof of Fermat's last theorem. In *On modular representations of ${\rm Gal}(\overline{\mathbb Q}/\mathbb Q)$ arising from modular forms* by Ken Ribet he cites Serre's (awesome) paper *Sur les représentations modulaires de degré $2$ de ${\rm Gal}(\overline{\mathbb Q}/\mathbb Q)$*, which (4.1.12) says this follows immediately from the theory of Tate curves.

It is pretty easy: the Tate curve gives you a explicit description the field obtained by adjoining the $\ell$-torsion points to $\mathbb Q_p$, and one can just check directly that the divisibility condition implies that this field (and thus the mod $\ell$ representation on the $\ell$-torsion points) is unramified at $p$.

Nonetheless I'm curious to know if anyone writes this down explicitly anywhere in the literature.