Supersingular j-invariants and the class number of Q(√-p)

Let p > 3 be a prime. Let D be the discriminant of an imaginary quadratic order. Assume that vertical bar D vertical bar < 4 root p/3 and (D/p) = - 1. We compute the number of F-p-roots of the class polynomials H-D(X). Suppose Q(root D-1) not equal Q(root D-2), we prove that two class polynomials H-D1 (X) and H-D2 (X) have a common root in F-p if and only if D-1 D-2 - 4p is a perfect square. Furthermore, any three class polynomials do not have a common root in F-p. As an application, we propose a deterministic algorithm for computing the class number of Q(root-p).