Let $K$ be a quadratic imaginary field, and E an elliptic curve whose endomorphism ring is isomorphic to the full ring of integers of K. Let j be its j-invariant, and c an integral ideal of K. Consider the following tower:

K(j,E[c]) / K(j,h(E[c])) / K(j) / K,

where h here is any Weber function on E. (Note that K(j) is the Hilbert class field of K).

We know that all these extensions are Galois, and any field has ABELIAN galois group over any smaller field, EXCEPT POSSIBLY THE BIGGEST ONE (namely, K(j,E[c]) / K).

Questions:

- Does the biggest one have to be abelian? Give a proof or counterexample.

My suspicion: No, it doesn't. I've been trying an example with K = Q($\sqrt{-15}$), E = C/O_K, and c = 3; it just requires me to factorise a quartic polynomial over Q-bar, which SAGE apparently can't do.

- What about if I replace E[c] in the above by E_tors, the full torsion group?