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

Guanju Xiao, Lixia Luo, Yingpu Deng

Published 2021-01-13Version 1

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 $\tilde{O}(p^{1/2})$ if we use probabilistic factorization algorithms and assume the Generalized Riemann Hypothesis.