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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?

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  • $\begingroup$ 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. $\endgroup$ – Asvin Jan 30 '19 at 14:01

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