Computation of Relative Class Numbers of Imaginary Abelian Number Fields

Abstract

We develop an efficient technique for computing relative class numbers of imaginary abelian fields, efficient enough to enable us to easily compute relative class numbers of imaginary cyclic fields of degrees 32 and conductors greater than 1013, or of degrees 4 and conductors greater than 1015. Acccording to our extensive computation, all the 166204 imaginary cyclic quartic fields of prime conductors p less than 107 have relative class numbers less than p/2. Our major innovation is a technique for computing numerically root numbers appearing in some functional equations.