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

Abstract

We use a variant of Vinogradov’s method to show that the density of the set of prime numbers $${p \equiv -1 {\rm mod} 4}$$ 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.