Supersingular $j$-invariants and the Class Number of $\mathbb{Q}(\sqrt{-p})$

Abstract

For a prime $p>3$, let $D$ be the discriminant of an imaginary quadratic order with $|D|<\frac{4}{\sqrt{3}}\sqrt{p}$. We research the solutions of the class polynomial $H_D(X)$ mod $p$ in ${\mathbb{F}}_p$ if $D$ is not a quadratic residue in ${\mathbb{F}}_p$. We also discuss the common roots of different class polynomials in ${\mathbb{F}}_p$. As a result, we get a deterministic algorithm (Algorithm 3) for computing the class number of $\mathbb{Q}(\sqrt{-p})$. The time complexity of Algorithm 3 is $O(p^{3/4+\epsilon })$.