Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Computing isomorphisms between lattices
HTML articles powered by AMS MathViewer

by Tommy Hofmann and Henri Johnston HTML | PDF
Math. Comp. 89 (2020), 2931-2963 Request permission


Let $K$ be a number field, let $A$ be a finite-dimensional semisimple $K$-algebra, and let $\Lambda$ be an $\mathcal {O}_{K}$-order in $A$. It was shown in previous work that, under certain hypotheses on $A$, there exists an algorithm that for a given (left) $\Lambda$-lattice $X$ either computes a free basis of $X$ over $\Lambda$ or shows that $X$ is not free over $\Lambda$. In the present article, we generalize this by showing that, under weaker hypotheses on $A$, there exists an algorithm that for two given $\Lambda$-lattices $X$ and $Y$ either computes an isomorphism $X \rightarrow Y$ or determines that $X$ and $Y$ are not isomorphic. The algorithm is implemented in Magma for $A=\mathbb {Q}[G]$, $\Lambda =\mathbb {Z}[G]$, and $\Lambda$-lattices $X$ and $Y$ contained in $\mathbb {Q}[G]$, where $G$ is a finite group satisfying certain hypotheses. This is used to investigate the Galois module structure of rings of integers and ambiguous ideals of tamely ramified Galois extensions of $\mathbb {Q}$ with Galois group isomorphic to $Q_{8} \times C_{2}$, the direct product of the quaternion group of order $8$ and the cyclic group of order $2$.
Similar Articles
  • Retrieve articles in Mathematics of Computation with MSC (2010): 11R33, 11Y40, 16Z05
  • Retrieve articles in all journals with MSC (2010): 11R33, 11Y40, 16Z05
Additional Information
  • Tommy Hofmann
  • Affiliation: Fachbereich Mathematik, Technische Universität Kaiserslautern, 67663 Kaiserslautern, Germany
  • MR Author ID: 1074375
  • Email:
  • Henri Johnston
  • Affiliation: Department of Mathematics, University of Exeter, Exeter, EX4 4QF United Kingdom
  • MR Author ID: 776746
  • ORCID: 0000-0001-5764-0840
  • Email:
  • Received by editor(s): March 2, 2019
  • Received by editor(s) in revised form: January 13, 2020
  • Published electronically: June 1, 2020
  • Additional Notes: The first author was supported by Project II.2 of SFB-TRR 195 “Symbolic Tools in Mathematics and their Application” of the German Research Foundation (DFG)
    The second author was supported by EPSRC First Grant EP/N005716/1 “Equivariant Conjectures in Arithmetic”.
  • © Copyright 2020 American Mathematical Society
  • Journal: Math. Comp. 89 (2020), 2931-2963
  • MSC (2010): Primary 11R33, 11Y40, 16Z05
  • DOI:
  • MathSciNet review: 4136552