Let $E/\mathbb{Q}$ be an elliptic curve and $p$ a prime such that $E$ has ordinary redcution at p. Further, let $$\rho_{E,p}:{\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to {\rm GL}_2(\mathbb{Z}/p\mathbb{Z})$$ be the mod $p$ Galois representation associated to $E$ and let $\sigma_p\in{\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ be the $p$-Frobenius.
What can be said about the matrix $A_p =\rho_{E,p}(\sigma_p)$?
Can anything be said about $A_p = \rho_{E,p}(\sigma_p)$ if $p$ is a prime of bad reduction?
The representation $\rho_{E, p}$ is always ramified at $p$; it has to be, because its determinant is the mod $p$ cyclotomic character. Thus $\rho_{E, p}(\sigma_p)$ is not well-defined, as the conjugacy class of "$p$-Frobenius elements" is only defined modulo the inertia subgroup at $p$.