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arXiv · 2101.04937

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

Guanju Xiao·Lixia Luo·Yingpu Deng
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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 $O(p^{3/4+\epsilon})$.

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Guanju Xiao, Lixia Luo, Yingpu Deng. 2021-01-13. Supersingular $j$-invariants and the Class Number of $\mathbb{Q}(\sqrt{-p})$. https://arxiv.org/abs/2101.04937

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