article published Markus Kirschmer author 82258 John Voight author 102 department 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. Society for Industrial & Applied Mathematics (SIAM)2010 eng General MathematicsGeneral Computer Science 0097-5397 1095-711110.1137/080734467 3951714-1747 yes Kirschmer, Markus, and John Voight. “Algorithmic Enumeration of Ideal Classes for Quaternion Orders.” <i>SIAM Journal on Computing</i> 39, no. 5 (2010): 1714–47. <a href="https://doi.org/10.1137/080734467">https://doi.org/10.1137/080734467</a>. M. Kirschmer, J. Voight, SIAM Journal on Computing 39 (2010) 1714–1747. M. Kirschmer and J. Voight, “Algorithmic Enumeration of Ideal Classes for Quaternion Orders,” <i>SIAM Journal on Computing</i>, vol. 39, no. 5, pp. 1714–1747, 2010, doi: <a href="https://doi.org/10.1137/080734467">10.1137/080734467</a>. Kirschmer, M., &#38; Voight, J. (2010). Algorithmic Enumeration of Ideal Classes for Quaternion Orders. <i>SIAM Journal on Computing</i>, <i>39</i>(5), 1714–1747. <a href="https://doi.org/10.1137/080734467">https://doi.org/10.1137/080734467</a> @article{Kirschmer_Voight_2010, title={Algorithmic Enumeration of Ideal Classes for Quaternion Orders}, volume={39}, DOI={<a href="https://doi.org/10.1137/080734467">10.1137/080734467</a>}, number={5}, journal={SIAM Journal on Computing}, publisher={Society for Industrial &#38; Applied Mathematics (SIAM)}, author={Kirschmer, Markus and Voight, John}, year={2010}, pages={1714–1747} } Kirschmer M, Voight J. Algorithmic Enumeration of Ideal Classes for Quaternion Orders. <i>SIAM Journal on Computing</i>. 2010;39(5):1714-1747. doi:<a href="https://doi.org/10.1137/080734467">10.1137/080734467</a> Kirschmer, Markus, and John Voight. “Algorithmic Enumeration of Ideal Classes for Quaternion Orders.” <i>SIAM Journal on Computing</i>, vol. 39, no. 5, Society for Industrial &#38; Applied Mathematics (SIAM), 2010, pp. 1714–47, doi:<a href="https://doi.org/10.1137/080734467">10.1137/080734467</a>. 428032023-03-07T08:49:35Z2023-04-04T09:25:08Z