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!



Extensions obtained adding torsion points of an elliptic curve
This question is discussed very carefully in Section 3 of the paper *Mod $p$ representations
on elliptic curves*, by Frank Calegari (available [here](http://www.math.northwestern.edu/~fcale)). 


In particular, after noting that the answer is positive when $p \leq 5$ (as was already
observed in the comments above), he proves (in Theorems 3.3 and 3.4) that if 
$p \geq 7$ then there exists a continuous representation $\rho: Gal(\overline{\mathbb Q}/
\mathbb Q) \to GL\_2(\mathbb F\_p)$ with cyclotomic determinant which *does not* come from
the $p$-torsion of an elliptic curve. (He also notes that the same result was established
by Dieulefait, in the paper *Existence of non-elliptic mod $\ell$ Galois representations for every $\ell > 5$*.)


