Orbits in the Leech lattice

We provide an algorithm for determining whether two vectors in the Leech lattice are equivalent under its isometry group, the Conway group Coo of order similar to 8 x 10(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 similar to 24 with good properties such as having small determinant. Our methods rely on and develop the work of R. T. Curtis.