Genus of a quadratic form

In mathematics, the genus is a classification of quadratic forms and lattices over the ring of integers.

An integral quadratic form is a quadratic form on Zn, or equivalently a free Z-module of finite rank.

The Smith–Minkowski–Siegel mass formula gives the weight or mass of the quadratic forms in a genus, the count of equivalence classes weighted by the reciprocals of the orders of their automorphism groups.

For binary quadratic forms there is a group structure on the set C of equivalence classes of forms with given discriminant.

The principal genus, the genus containing the principal form, is precisely the subgroup C2 and the genera are the cosets of C2: so in this case all genera contain the same number of classes of forms.