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