On the $16$-rank of class groups of $\mathbb{Q}(\sqrt{-8p})$ for $p\equiv -1\bmod 4$

Abstract

We use a variant of Vinogradov's method to show that the density of the set of prime numbers $p\equiv -1\bmod~4$ for which the class group of the imaginary quadratic number field $\mathbb{Q}(\sqrt{-8p})$ has an element of order $16$ is equal to $1/16$, as predicted by the Cohen-Lenstra heuristics.