In this thesis we look at particular details of class field theory for complex multiplication fields. We begin by giving some background on fields, abelian varieties, and complex multiplication. We then turn to the first task of this thesis and give an implementation in Sage of a classical algorithm to compute the Hilbert class field of a quadratic complex multiplication field using the j-invariant of elliptic curves with complex multiplication by the ring of integers of the field, and we include three explicit examples to illustrate the algorithm. The second part of this thesis contains new results: Let K be a sextic complex multiplication field with Galois closure L such that the Galois group of L over Q is isomorphic to D12, the dihedral group with twelve elements. For each complex multiplication type Phi of K, we compute the reflex field and reflex type of the pair (K, Phi) explicitly. We then illustrate our results with the case of K = Q[x]/(x6 - 2x5 +2x4 +2x3 +4x2 -4x+2).