Let $E/\mathbb{Q}$ be an elliptic curve. For a prime $p$, we have the mod $p$ representation \begin{equation*} \bar{\rho}_{E,p}: G_{\bar{\mathbb{Q}}/\mathbb{Q}} \rightarrow Aut(E[p]) \end{equation*} and the $p$-adic representation \begin{equation*} \rho_{E,p}: G_{\bar{\mathbb{Q}}/\mathbb{Q}} \rightarrow Aut(T_pE) \end{equation*} where $T_pE$ is the p-adic Tate module.
We know that by Serre, 1972, that if $E$ has good reduction at $p$, and $p\geq 5$, then $\bar{\rho}_{E,p}$ is surjective if and only if $\rho_{E,p}$ is surjective. However, I have seen cases where it is stated that as long as $p\geq 5$, then this relation holds, without any conditions on the reduction at $p$ of E. Is this true?
You can find a more general version of this implication, with a detailed proof, in the first few pages of Swinnerton-Dyer's article in the Antwerp III proceedings (Springer Lecture Notes vol. 331). Swinnerton-Dyer certainly does not make any assumptions on the reduction type at $p$.