Elliptic curves with complex multiplication and abelian division fields

Let \(K\) be an imaginary quadratic field, and let \(\mathcal{O}_{K,f}\) be an order in \(K\) of conductor \(f\geq 1\). Let \(E\) be an elliptic curve with CM by \(\mathcal{O}_{K,f}\), such that \(E\) is defined by a model over \(\mathbb{Q}(j_{K,f})\), where \(j_{K,f}=j(E)\). In this article, we classify the values of \(N\geq 2\) and the elliptic curves \(E\) such that (i) the division field \(\mathbb{Q}(j_{K,f},E[N])\) is an abelian extension of \(\mathbb{Q}(j_{K,f})\), and (ii) the \(N\)-division field coincides with the \(N\)-th cyclotomic extension of the base field.