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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?

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  • $\begingroup$ 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. $\endgroup$ – debanjana Jan 12 at 23:45

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