A refined Malle conjecture for Heisenberg groups

Abstract

Based on a conjecture of Loughran and the second author, we give an explicit prediction for the leading constant in Malle's conjecture for Galois $\mc{H}$-extensions of $\Q$ ordered by discriminant, where $\mc{H}$ is the $3\times 3$ Heisenberg group over $\F_4$. The predicted leading constant is not a single Euler product, but rather a sum of two distinct Euler products. Our methods also give an efficient algorithm for computing the conjectural Loughran--Santens leading constant for many $2$-groups of nilpotency class $2$.