# Galois action on CM elliptic curves

Let $E$ be an elliptic curve $E$ defined over a number field $K$ such that $E$ has complex multiplication by the maximal order in the ring of integers of an imaginary quadratic field $F$. Let $K^{cycl}$ be the extension of $K$ obtained by adjoining all roots of unity to $K$, and $K(E_{tors})$ the field obtained by adjoining all coordinates of torsion points of $E$ to $K$. Then what is $Gal(K(E_{tors}) / K^{cycl})$?

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