Imaginary quadratic number fields with class groups of small exponent

TL;DR

This paper computes all imaginary quadratic fields with discriminants up to 3.1×10^{20} where the class group exponent divides 8, assuming no Siegel zeros, extending previous classifications.

Contribution

It provides a comprehensive computational classification of imaginary quadratic fields with small class group exponents under certain assumptions.

Findings

All such discriminants with |D| ≤ 3.1×10^{20} are identified.

Class group exponents dividing 8 are characterized for these fields.

The results depend on the assumption that no Siegel zeros exist.

Abstract

Let $D<0$ be a fundamental discriminant and denote by $E(D)$ the exponent of the ideal class group $\text{Cl}(D)$ of $K={\mathbb Q}(\sqrt{D})$. Under the assumption that no Siegel zeros exist we compute all such $D$ with $E(D)$ is a divisor of $8$. We compute all $D$ with $|D|\leq 3.1\cdot 10^{20}$ such that $E(D)\leq 8$.