Let $E$ be an elliptic curve with CM by an order in the imaginary quadratic field $K$. Is there some easy way how to prove that the extension $K(j(E))/\mathbb Q$ is abelian?

**Update**

In general, the extension $K(j(E))/\mathbb Q$ is not abelian. The question therefore is, whether there is an elementary way to see that $K(j(E))/K$ is abelian (or even Galois).

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