## Experimental Mathematics

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

### 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 $10^{13}$, 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