The Conway–Sloane tetralattice pairs are non-isometric

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 104. In this paper, we use our theory of lattice invariants to prove this conjecture.