I suppose this question could be phrased in terms of Galois representations, but I'm asking it this way.


Let $n>1$ be an integer. If $K$ is a number field with $\operatorname{Gal}(K/\mathbb{Q}) \cong GL\_2(\mathbb{Z}/n\mathbb{Z})$ (edit) and containing the $n$-th roots of unity (/edit), must there exist an elliptic curve $E$ defined over $\mathbb{Q}$ such that $K \subseteq \mathbb{Q}(E[n])$? It's not necessary to produce the curve, although it would be cool.


If not, is there a good way to test if this holds for a given $K$?


(If you want to restrict to the case where $n$ is a prime power, that's fine.)



Is every $GL_2(\mathbb{Z}/n\mathbb{Z})$-extension contained in some elliptic curve's torsion field?
I have two things to add to this discussion.


$\bullet$ For $n = 2$ and $n = 3$, every Galois extension of $\mathbb{Q}$ with Galois group ${\rm GL}\_{2}(\mathbb{Z}/n\mathbb{Z})$ *does* arise from an elliptic curve (by a result of Shepard-Barron and Taylor from 1997 - see the reference in the paper of Dieulefait linked to below.) For $n = 4$, this is not true (see [this paper](http://www.ams.org/journals/proc/2008-136-01/S0002-9939-07-08899-5/)).


$\bullet$ For $p \geq 7$ prime, it is known that not every Galois representation $\rho : G\_{\mathbb{Q}} \to GL\_{2}(\mathbb{F}\_{p})$ arises from an elliptic curve, even if we restrict $\rho$ to have cyclotomic determinant. This is shown in 
[a paper of Dieulefait](http://www2.math.ethz.ch/EMIS/journals/EM/expmath/volumes/13/13.3/Dieulefait.pdf) for $p \geq 7$, because one can construct Galois representations using modular forms of different weights. This somehow doesn't seem quite the same as starting with a field $K/\mathbb{Q}$ with $Gal(K/\mathbb{Q}) \cong GL\_{2}(\mathbb{F}\_{p})$. (Any two isomorphisms between $Gal(K/\mathbb{Q})$ and $GL\_{2}(\mathbb{F}\_{p})$ differ by an automorphism of $GL\_{2}(\mathbb{F}\_{p})$ and I haven't been able to find a convenient reference for what ${\rm Aut}(GL\_{2}(\mathbb{F}\_{p}))$ is.)


