For an elliptic curve E defined over a number field K, the heuristic density of the set of primes of K for which E has cyclic reduction is given by an inclusion–exclusion sum delta _{E/K} involving the degrees of the m-division fields K_m of E over K. This density can be proved to be correct under assumption of GRH. For E without complex multiplication (CM), we show that delta _{E/K} is the product of an explicit non-negative rational number reflecting the finite entanglement of the division fields of E and a universal infinite Artin-type product. For E admitting CM over K by a quadratic order mathcal {O}, we show that delta _{E/K} admits a similar ‘factorization’ in which the Artin type product also depends on mathcal {O}. For E admitting CM over overline{K} by an order mathcal {O}not subset K, which occurs for K=textbf{Q}, the entanglement of division fields over K is non-finite. In this case we write delta _{E/K} as the sum of two contributions coming from the primes of K that are split and inert in mathcal {O}. The split contribution can be dealt with by the previous methods, the inert contribution is of a different nature. We determine the ways in which the density can vanish, and provide numerical examples of the different kinds of densities.