 
Summary: ORBITS IN THE LEECH LATTICE
DANIEL ALLCOCK
Abstract. We provide an algorithm for determining whether two
vectors in the Leech lattice are equivalent under its isometry group,
the Conway group Co 0 of order # 8 × 10 18 . Our methods rely on
and develop the work of R. T. Curtis, and we describe our in
tended applications to the symmetry groups of Lorentzian lattices
and the enumeration of lattices of dimension # 24 with good prop
erties such as having small determinant. Our algorithm reduces
the test of equivalence to # 4 tests under the subgroup 2 12 :M 24 ,
and a test under this subgroup to # 12 tests under M 24 . We also
give algorithms for testing equivalence under these two subgroups.
Finally, we analyze the performance of the algorithm.
1. Introduction
The Leech lattice # is a lattice in 24dimensional Euclidean space
with many remarkable properties, for us the most important of which
is that its isometry group (modulo {±I}) is one of the sporadic finite
simple groups. The isometry group is called the Conway group Co 0 ,
and our purpose is to present an algorithm for a computer to determine
whether two given vectors of # are equivalent under Co 0 . The group
