genera

Consider h_+(d) proper equivalence classes of forms with discriminant d equal to the field discriminant, then they can be subdivided equally into 2^(r - 1) genera of h_+(d)/2^(r - 1) forms which form a subgroup of the proper equivalence class group under composition, where r is the number of distinct prime divisors of d. This theorem was proved by Gauss in 1801.

form genus | genus theorem