# The Conway-Sloane tetralattice pairs are non-isometric

## TL;DR

This paper proves that all pairs of isospectral lattices in Conway and Sloane's 4-parameter family are non-isometric when parameters differ, confirming a conjecture using new lattice invariants.

## Contribution

The paper introduces a new theory of lattice invariants that proves Conway and Sloane's conjecture for their family of isospectral lattices.

## Key findings

- All pairs in the family are non-isometric when parameters differ.
- The conjecture is verified for classical integral lattices of determinant up to 10^4.
- The new invariants distinguish non-isometric lattices in the family.

## Abstract

Conway and Sloane constructed a 4-parameter family of pairs of isospectral lattices of rank four. They conjectured that all pairs in their family are non-isometric, whenever the parameters are pairwise different, and verified this for classical integral lattices of determinant up to $10^4$. In this paper, we use our theory of lattice invariants to prove this conjecture.