# Is $p$ an anomalous prime?

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?

• If $m$ is an integer such that $v(m)=0$, then we know (for example Silverman VIII.1.4) that $E(K)[m]\hookrightarrow \tilde{E}_v(k_v)$, but here $v\mid p$ and the situation isn't totally clear to me. – debanjana Jan 12 at 23:45