The image of the adelic Galois representation of an elliptic curve with complex multiplication

Let $E\mathrm{/}\mathbb{Q}$ be an elliptic curve and let ${\rho }_E\text{\hspace{0.17em}⁣}:\mathrm{Gal}(\bar{\mathbb{Q}}\mathrm{/}\mathbb{Q})\to \mathrm{GL}(2,\widehat{\mathbb{Z}})$ be the adelic Galois representation attached to $E$. We describe and implement an algorithm to compute the image of ${\rho }_E$ in $\mathrm{GL}(2,\widehat{\mathbb{Z}})$ (up to conjugation) for an elliptic curve $E\mathrm{/}\mathbb{Q}$ with complex multiplication (CM) and $j$-invariant not $0$ or $1728$. In the process, we prove certain entanglement results between division fields of elliptic curves over $\mathbb{Q}$ with CM.