When adding to the rational the $p$-torsion points $E[p]$ of an elliptic curve we obtain an extension containing the $p$-th roots of the unity, and whose Galois group can be embedded in $GL(2, \mathbb{F}_p)$. To what extent are such extensions coming from elliptic curves?

I mean, assume $K/\mathbb{Q}$ to be an extension whose Galois group can be embedded in $GL(2, \mathbb{F}_p)$ and containing the $p$-th roots of the unity (which is required to expect a positive answer), is $K$ obtained adding to $\mathbb{Q}$ the torsion points of some elliptic curve defined over the rationals?

Note that I'm not considering a particular Galois representation, but just Galois groups that can be embedded in some way into $GL(2, \mathbb{F}_p)$. Thanks!

Propriétés des points d'ordre fini des courbes elliptiques, Inventiones 1972. $\endgroup$Modular curves and the Eisenstein ideal, Publ. Math. IHES 1977. $\endgroup$5more comments