How big is the image of the Galois representations attached to CM elliptic curves?

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Abstract
Using an analogue of Serre’s open image theorem for elliptic curves with complex multiplication, one can associate to each CM elliptic curve E E defined over a number field F F a natural number I ( E / F ) \mathcal {I}(E/F) which describes how big the image of the Galois representation associated to E E is. We show how one can compute I ( E / F ) \mathcal {I}(E/F) , using a closed formula that we obtain from the classical theory of complex multiplication.