COMPUTATION OF THE RELATIVE CLASS-NUMBERS OF SOME HILBERT CLASS FIELDS

We give an expression for the relative class number of any CM-field that is suitable for numerical computation. This expression is then used in order to compute relative class numbers of 2-narrow Hilbert class fields of real quadratic fields for which their 2-Sylow subgroups of their ideal class groups are cyclic. These computations suggest that there are exactly 24 such 2-narrow Hilbert class fields with relative class number one, and that they have degree 8 or 16.