Computing ideal classes representatives in quaternion algebras

PACETTI, ARIEL; SIROLLI, NICOLÁS

MATHEMATICS OF COMPUTATION

AMER MATHEMATICAL SOC

Lugar: Providence; Año: 2014 vol. 83 p. 2479 - 2507

0025-5718

Let $K$ be a totally real number field and let $B$ be a totally definitequaternion algebra over $K$. In this article, given a set of representatives forideal classes for a maximal order in $B$, we show how to construct in anefficient 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 improvesthe equivalence checking process by using a simple calculation with globalunits. As an application, we compute ideal classes representatives for an orderof level $30$ in an algebra over the real quadratic field $Q[sqrt{5}]$.