Supersingular j-invariants and the class number of ℚ(−p)
Guanju Xiao, Lixia Luo, Yingpu Deng · International Journal of Number Theory · 2021
Let [Formula: see text] be a prime. Let [Formula: see text] be the discriminant of an imaginary quadratic order. Assume that [Formula: see text] and [Formula: see text]. We compute the number of [Formula: see text]-roots of the class polynomials [Formula: see text]. Suppose [Formula: see text], we prove that two class polynomials [Formula: see text] and [Formula: see text] have a common root in [Formula: see text] if and only if [Formula: see text] is a perfect square. Furthermore, any three class polynomials do not have a common root in [Formula: see text]. As an application, we propose a deterministic algorithm for computing the class number of [Formula: see text].