Experimental Mathematics

Computation of relative class numbers of imaginary abelian number fields

Stéphane Louboutin

Full-text: Open access

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 $10^{13}$, or of degrees $4$ and conductors greater than $10^{15}$. Acccording to our extensive computation, all the $166204$ imaginary cyclic quartic fields of prime conductors $p$ less than $10^7$ have relative class numbers less than $p$/2. Our major innovation is a technique for computing numerically root numbers appearing in some functional equations.

Article information

Source
Experiment. Math., Volume 7, Issue 4 (1998), 293-303.

Dates
First available in Project Euclid: 14 March 2003

Permanent link to this document
https://projecteuclid.org/euclid.em/1047674147

Mathematical Reviews number (MathSciNet)
MR1678103

Zentralblatt MATH identifier
0929.11065

Subjects
Primary: 11Y40: Algebraic number theory computations
Secondary: 11M20: Real zeros of $L(s, \chi)$; results on $L(1, \chi)$ 11R20: Other abelian and metabelian extensions 11R29: Class numbers, class groups, discriminants

Keywords
Imaginary abelian number field relative class number

Citation

Louboutin, Stéphane. Computation of relative class numbers of imaginary abelian number fields. Experiment. Math. 7 (1998), no. 4, 293--303. https://projecteuclid.org/euclid.em/1047674147


Export citation