Computing ideal classes representatives in quaternion algebras

Abstract

Let K be a totally real number field and let B be a totally definite quaternion algebra over K. Given a set of representatives for ideal classes for a maximal order in B, we show how to construct in an efficient way a set of representatives of ideal classes for any Bass order in B. The algorithm does not require any knowledge of class numbers, and improves the equivalence checking process by using a simple calculation with global units. As an application, we compute ideal classes representatives for an order of discriminant 30 in an algebra over the real quadratic field Q ([√ 5].