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I found the following claim - without reference - in the (famous) book ''Modular Forms and Fermat’s Last Theorem'':

Let $E$ be a semistable elliptic curve over $\mathbb{Q}$. Let $\Delta_E$ be the minimal discriminant. Then the residual representation $\bar{\rho}_{E,p}$ has the following properties:

  • If $\ell\neq p$, then $\bar{\rho}_{E,p}$ is unramified at $\ell$ iff $p\mid\mathrm{ord}_\ell(\Delta_E)$.
  • $\bar{\rho}_{E,p}$ is flat at $p$ iff $p\mid\mathrm{ord}_p(\Delta_E)$.

I want to know references for this result. Or, if it can be shown with simple argument, any hint would also be helpful.

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