Algorithmic enumeration of ideal classes for quaternion orders

TL;DR

This paper introduces algorithms for counting and enumerating ideal classes in quaternion algebra orders, analyzing their efficiency and applying them to classify orders with small class numbers.

Contribution

It presents new algorithms for ideal class enumeration in quaternion orders and explores related computational problems in algebraic number theory.

Findings

Algorithms successfully enumerate ideal classes and compute related structures.

Complete list of definite Eichler orders with class number ≤ 2 provided.

Analysis of algorithm runtime and complexity included.

Abstract

We provide algorithms to count and enumerate representatives of the (right) ideal classes of an Eichler order in a quaternion algebra defined over a number field. We analyze the run time of these algorithms and consider several related problems, including the computation of two-sided ideal classes, isomorphism classes of orders, connecting ideals for orders, and ideal principalization. We conclude by giving the complete list of definite Eichler orders with class number at most 2.