Malle's conjecture with multiple invariants

We define invariants ${\mathrm{inv}}_1,\dots ,{\mathrm{inv}}_m$ of Galois extensions of number fields with a fixed Galois group. Then, we propose a heuristic in the spirit of Malle’s conjecture which asymptotically predicts the number of extensions that satisfy ${\mathrm{inv}}_i\le X_i$ for all $X_i$. The resulting conjecture is proved for abelian Galois groups. We also describe refined Artin conductors that carry essentially the same information as the invariants ${\mathrm{inv}}_1,\dots ,{\mathrm{inv}}_m$.