# Implementation for computing Shintani domains

By "Shintani domain", I mean a fundamental domain for the action of the totally positive units of a totally real number field k with $[k \colon \mathbb{Q}]=n$ (or more generally, those congruent to 1 modulo some integral ideal of k) on $(\mathbb{R}^{+} )^n$. Shintani showed that there exists such a domain which is a finite disjoint union of simplicial cones.

After combing through the literature, I have found complete descriptions of these domains for quadratic and cubic fields. Other than that, I have found two papers with algorithms that describe how this domain can be found:

R. Okazaki, "On an effective determination of a Shintani's decomposition of the cone $\mathbb{R}^{+n}$", J. Math. Kyoto U. 33 (1993) 1057--1070

where only one example is given, namely, a Shintani domain for the quintic subfield of $Q(\zeta_{11})$, and

U. Halbritter and M. Pohst, "On lattice bases with special properties", J. de Théorie des Nombres de Bordeaux, 12 (2000), 437-453.

where only a partial example is given of the computation of a Shintani domain for a quartic field

Does anyone know of an implementation such an algorithm that has been or is being developed? Thank you for your help.

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