Single-Class Genera of Positive Integral Lattices

TL;DR

This paper classifies all positive definite primitive integer lattices in dimensions 4 and 5 that form a single isometry class genus, using computational methods based on the Smith-Minkowski-Siegel mass formula and Watson's mapping.

Contribution

It completes the classification of single-class genera in dimensions 4 and 5 and corrects previous classification errors in other dimensions.

Findings

Complete list of single-class primitive Z-lattices in dimensions 4 and 5.

Correction of previous classification mistakes.

Integration of results into the Catalogue of Lattices.

Abstract

We give an enumeration of all positive definite primitive Z-lattices in dimension >= 3 whose genus consists of a single isometry class. This is achieved by using bounds obtained from the Smith-Minkowski-Siegel mass formula to computationally construct the square-free determinant lattices with this property, and then repeatedly calculating pre-images under a mapping first introduced by G. L. Watson. We hereby complete the classification of single-class genera in dimensions 4 and 5 and correct some mistakes in Watson's classifications in other dimensions. A list of all single-class primitive Z-lattices has been compiled and incorporated into the Catalogue of Lattices.