Orbits in the Leech Lattice

Abstract

We provide an algorithm for determining whether two vectors in the Leech lattice are equivalent under its isometry group, the Conway group $\co0$ of order $\sim8\times10^{18}$. Our algorithm reduces the test of equivalence to at most four tests under the subgroup $2^{12}{:}M_{24}$ and a test under this subgroup to at most 12 tests under $M_{24}$. We also give algorithms for testing equivalence under these two subgroups. We describe our intended applications to the symmetry groups of Lorentzian lattices and the enumeration of lattices of dimension ${}\sim24$ with good properties such as having small determinant. Our methods rely on and develop the work of R. T. Curtis.