The lattice invariants are defined by
The lattice roots satisfy the cubic equation
and are denoted by . The discriminant (§1.11(ii)) is given by
Similarly for and . As functions of and , and are meromorphic and is entire.
Conversely, , , and the set are determined uniquely by the lattice independently of the choice of generators. However, given any pair of generators , of , and with defined by (23.2.1), we can identify the individually, via
In what follows, it will be assumed that (23.3.9) always applies.