I am seeking References for facts from chapter I of Cornell-Silverman-Stevens, "Modular Forms and Fermat's Last Theorem."
On page 6 of the text, the authors provide in Theorem 2.11 some "basic facts" about Galois representations attached to elliptic curves, with no references given. The first half of the theorem about nonreduced $l$-adic Galois representations attached to elliptic curves is proven in Chapter 9 of Diamond and Shurman's "A First Course in Modular Forms," while I cannot seem to find a reference for the second half of the theorem regarding the unramification/flatness of the reduced $l$-adic Galois representation attached an a elliptic curve. Where can I find this information?
For reference, here is the theorem I would like to see a proof of or be given references for:
Let $E/\mathbb Q$ a semistable elliptic curve with minimal discriminant $\Delta_m(E)$. For $p$ prime, let $\overline{\rho}_{E, p}: G_\mathbb Q \to \operatorname{GL}_2(\mathbb F_p)$ be the reduced $p$-adic representation attached to $E$. If $l \neq p$ is prime, then $\overline{\rho}_{E, p}$ is unramified at $l$ iff $p | \operatorname{ord}_l(\Delta_m(E))$. We also have $\overline{\rho}_{E, p}$ is flat at $p$ iff $p | \operatorname{ord}_p(\Delta_m(E))$.