Computing the ideal class monoid of an order

Abstract

There are well known algorithms to compute the class group of the maximal order ${\mathcal{O}}_K$ of a number field $K$ and the group of invertible ideal classes of a non-maximal order $R$. In this paper we explain how to compute also the isomorphism classes of non-invertible ideals of an order $R$ in a finite product of number fields $K$. In particular we also extend the above-mentioned algorithms to this more general setting. Moreover, we generalize a theorem of Latimer and MacDuffee providing a bijection between the conjugacy classes of integral matrices with given minimal and characteristic polynomials and the isomorphism classes of lattices in certain $\mathbb{Q}$-algebras, which under certain assumptions can be explicitly described in terms of ideal classes.

arXiv commentfinal version

Cited by in corpus (10)

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• On the lower bound of the number of abelian varieties over ${\mathbb{F}}_p$
• Modules over orders, conjugacy classes of integral matrices, and abelian varieties over finite fields
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• Abelian varieties over finite fields and their groups of rational points
• Computing base extensions of ordinary abelian varieties over finite fields
• Class number for pseudo-Anosovs