Let $K$ be a fixed number field and $E$ be any elliptic curve over $K$. When we adjoin to $K$ the $p$-torsion points $E[p]$, we obtain an extension whose Galois group can be embedded in $GL(2, \mathbb{F}_p)$.

Under what conditions (on $E$ and $p$) can we get that $K(E[p])/K$ is finite $p$-extension?