Cohen–Lenstra heuristic and roots of unity

We report on computational results indicating that the well-known Cohen–Lenstra–Martinet heuristic for class groups of number fields may fail in many situations. In particular, the underlying assumption that the frequency of groups is governed essentially by the reciprocal of the order of their automorphism groups, does not seem to be valid in those cases. The phenomenon is related to the presence of roots of unity in the base field or in intermediate fields. For all the examples considered, we propose alternative predictions which agree closely with the data, and which are inspired by results of Gerth.