Let $E/\mathbb{Q}$ be an elliptic curve with Complex Multiplication by the ring of integers of an imaginary quadratic field $K$. Let $p$ be an odd prime of good supersingular reduction. We know by Serre-Tate that there exists a finite extension over $K$ such that $E$ attains good reduction everywhere. Is it in fact true that $E$ attains good reduction everywhere over $K(E[p])$ where $E[p]$ denotes the $p$-division points of $E$?
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