Let $E$ be an elliptic curve defined over a number field $F$ with supersingular reduction at an odd prime $p$. Let $E[p]$ denotes the set of $p$-torsion points of $E$ over an algebraic closure of $F$.

Statement - $E[p]$ is always irreducible for an elliptic curve $E$ with supersingular reduction at $p$.

Question - Is the above statement true for arbitrary $F$ or only when $F=\mathbb{Q}$, the set of rational numbers $?$

References are welcome.