Algorithmic Enumeration of Ideal Classes for Quaternion Orders

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. Since the very first calculations of Gauss for imaginary quadratic fields, the problem of computing the class group of a number field F has seen broad interest. Due to the evident close association between the class number and regulator (embodied in the Dirichlet class number formula), one often computes the class group and unit group in tandem as follows. Problem (ClassUnitGroup(ZF )). Given the ring of integers ZF of a number field F, compute the class group Cl ZF and unit group Z∗ F . This problem ...

• Type: Journal
• Year: 2010
• Where: SIAMCOMP
• Authors: Markus Kirschmer, John Voight
• Keywords: Class Group, Imaginary Quadratic fields, Quadratic fields, SIAMCOMP 2010