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Computing class groups and unit groups in Magma

Andreas-Stephan Elsenhans, John Voight

TL;DR

This work analyzes how Magma (V2.29) computes class groups and unit groups of number fields, detailing regimes from formalizable to heuristic and the interplay between factor-base, analytic, and saturation techniques. It articulates the factor-base framework, analytic class-number evaluations (with GRH-based rigor and heuristic Euler products), and unit-group algorithms including saturation via reductions modulo primes and discrete-log computations. The paper also documents implementation specifics, bug fixes (notably saturation issues), parallelization efforts, and practical benchmarks, alongside discussions of error sources and potential future enhancements. Overall, it offers a practical, regime-aware blueprint for exact and probabilistic computations in computational algebraic number theory and highlights directions for formalization and speedups.

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.

Computing class groups and unit groups in Magma

TL;DR

This work analyzes how Magma (V2.29) computes class groups and unit groups of number fields, detailing regimes from formalizable to heuristic and the interplay between factor-base, analytic, and saturation techniques. It articulates the factor-base framework, analytic class-number evaluations (with GRH-based rigor and heuristic Euler products), and unit-group algorithms including saturation via reductions modulo primes and discrete-log computations. The paper also documents implementation specifics, bug fixes (notably saturation issues), parallelization efforts, and practical benchmarks, alongside discussions of error sources and potential future enhancements. Overall, it offers a practical, regime-aware blueprint for exact and probabilistic computations in computational algebraic number theory and highlights directions for formalization and speedups.

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.

Paper Structure

This paper contains 8 sections, 3 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. A bit of history
  3. Basic algorithms
  4. The factor base method: an overview
  5. Analytic approaches
  6. Unit group algorithms
  7. Finishing the class group
  8. Conclusions

Key Result

Theorem 4.11

Assuming GRH, the set generates $\mathop{\mathrm{Cl}}\nolimits {\mathcal{O}}_K$.

Theorems & Definitions (18)

  • Remark 3.2
  • Remark 4.5
  • Remark 4.9
  • Remark 4.10
  • Theorem 4.11: Bach
  • proof
  • Remark 4.14
  • Theorem 5.1: Belabas--Friedman, Bach
  • proof
  • Theorem 5.2
  • ...and 8 more