# New invariants for integral lattices

## TL;DR

This paper introduces new invariants for integral lattices based on hyperspheres passing through a specific number of lattice points, with computational insights for 2D lattices of class number one.

## Contribution

The paper presents novel lattice invariants derived from hypersphere properties and applies them to classify two-dimensional lattices with class number one.

## Key findings

- Existence of hyperspheres passing through exactly n lattice points for all n>0
- Introduction of new invariants based on these hyperspheres
- Computational results for 2D lattices of class number one

## Abstract

Let $\Lambda$ be any integral lattice in Euclidean space. It has been shown that for every integer $n>0$, there is a hypersphere that passes through exactly $n$ points of $\Lambda$. Using this result, we introduce new lattice invariants and give some computational results related to two-dimensional Euclidean lattices of class number one.