The classes of singular moduli in the generalized Jacobian

Here the first sum is taken over the finite set of orbits of SL2(Z) on the set QD of all positive definite binary quadratic forms of discriminant −D, z is the (orbit of the) unique root in the upper half plane, and u(z) is half the order of the stabilizer. The internal sum in the second expression is over the primitive positive definite binary quadratic forms of discriminant −d and zd is a root in the upper half plane. We have u(d) = 1 unless d = 3 or d = 4, when u(3) = 3 and u(4) = 2 respectively. There are h(d) = #Pic(Od) terms in the internal sum, where Od is the imaginary quadratic order of discriminant −d, so the degree of the divisor in the sum for Tr(D) is equal to H(D) = ∑ D=df h(d)/u(d), the Hurwitz class number.