  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 $O(p^{3/4+\epsilon})$.
