On the Number of Equivalence Classes of Binary Forms of given Degree and given Discriminant

∈ GL2(Z) such that G(X, Y ) = F (aX + bY, cX + dY ). Denote by D(F ) the discriminant of a binary form F , and by OF the invariant order of an irreducible binary form F . We recall the definition of the invariant order of F which is less familiar. Write F (X, Y ) = a0X r + a1X Y + · · · + arY r and let θF be a zero of F (X, 1). Then OF is defined to be the Z-module with basis 1, a0θF , a0θ 2 F + a1θF , a0θ 3 F + a1θ 2 F + a2θF ,. . ., a0θ r−1 F + a1θ r−2 F + · · · + ar−2θF ; this is indeed an order, i.e., closed under multiplication. It is well-known that two equivalent binary forms have the same discriminant. Further, two equivalent irreducible binary forms have the same invariant order. The discriminant D(OF ) of OF is equal to D(F ) (see [8], [9] for a verification of these facts). Consequently, if K = Q(θF ),