Computing Class Groups and Unit Groups in Magma

Andreas-Stephan Elsenhans; John Voight

doi:10.1145/3815436.3815477

July 9, 2026Open Access

Computing Class Groups and Unit Groups in Magma

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Key Points

The aim is to compute class groups and unit groups of number fields using algorithms implemented in Magma.
Review main algorithms based on factor bases, relation collection, and analytic class number evaluation.
Distinguish behaviors across formalizable, rigorous, GRH-conditional, and heuristic regimes.
The algorithms show varied performance depending on the computational regime chosen.
Effective computation of class groups and unit groups can be achieved using these distinct approaches.

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Implication

Demonstrates algorithms to compute class and unit groups in number fields, indicating the effectiveness in various regimes.

Abstract

We describe the computation of class groups and unit groups of number fields as implemented in Magma (V2.29). After quickly reviewing the main algorithms based on factor bases, relation collection, and analytic class number evaluation, we distinguish their behavior across formalizable, rigorous, GRH-conditional, and heuristic regimes.

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Elsenhans et al. (2026) studied this question.

synapsesocial.com/papers/6a532e014f7abc118adec8f2 https://doi.org/10.1145/3815436.3815477

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Computing Class Groups and Unit Groups in Magma

Key Points

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Implication

Abstract

Discussion

Cite This Study

Computing Class Groups and Unit Groups in Magma

Key Points

Ask about this paper

Implication

Abstract

Discussion

Cite This Study

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