Is $p$ an anomalous prime?

Question

Let $E$ be an elliptic curve defined over the number field $K$ and $p$ be a rational prime such that for all $v\mid p$, $E$ has good ordinary reduction. If $E[p]\subseteq E(K)$, can we conclude $p$ is anomalous ie $p$ divides the order of $|\tilde{E}_v(k_v)|$ where $k_v$ is the residue field?