2026/07/07 by Jack B. Miller, Tim Santens
#math.NT
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 H-extensions of ℚ ordered by discriminant, where H is the 3× 3 Heisenberg group over \mathbbF4. 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.