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?