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?

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).

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

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?