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

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