# Galois representation of an elliptic curve with CM

Let $$E$$ be an elliptic curve with complex multiplication by an order $$\mathcal O$$ in an imaginary quadratic field $$K$$. Suppose that $$E$$ is defined over $$\mathbb Q(j(\mathcal O))$$. Let $$n$$ be an integer.

The Galois group $$G_{\mathbb Q(j(\mathcal O))}$$ acts on the torsion points $$E[n]$$. When is the image of the associated representation $$\rho_{n}\colon G_{\mathbb Q(j(\mathcal O))}\rightarrow GL_2(\mathbb Z/n\mathbb Z)$$ an abelian group?

• mathoverflow.net/q/134921, if I am not mistaken, this seems to be a duplicate of that question. The answer is that the image is abelian precisely when Q(j(O)) contains K. – Asvin Jan 30 '19 at 14:01