Computing ideal classes representatives in quaternion algebras
Ariel Pacetti, Nicolás Sirolli · Mathematics of Computation · 2014
Let K K be a totally real number field and let B B be a totally definite quaternion algebra over K K . Given a set of representatives for ideal classes for a maximal order in B B , we show how to construct in an efficient way a set of representatives of ideal classes for any Bass order in B 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 30 in an algebra over the real quadratic field Q [ 5 ] \mathbb {Q}[\sqrt {5}] .