Let $K$ be a finite extension of $\mathbb{Q}_p$ and $E/K$ an elliptic curve with good reduction. Does the $\mathbb{F}_p[\mathrm{Gal}(\overline{K})]$-module $E[p](\overline{K})$ determine whether the reduction is ordinary or supersingular? I guess this is true when $K=\mathbb{Q}_p$ because in the ordinary case the representation is reducible, while in the supersingular case, the representation is irreducible.
In general I know that the representation is reducible in the ordinary case because it has a $1$-dimensional unramified quotient. But I'm not sure whether or not the representation is irreducible in the supersingular case for arbitrary $K$.
The reason I ask this question is because I was wondering whether or not two curves over a number field with good reduction above $p$ that have isomorphic $p$-torsion representations necessarily have the same reduction type at primes above $p$ (ordinary or supersingular, that is).