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.

Keywords:

• Im aginary abelian number field
• relative class number