  We define invariants $\operatorname{inv}_1,\dots,\operatorname{inv}_m$ 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 $\operatorname{inv}_i\leq X_i$ 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 $\operatorname{inv}_1,\dots,\operatorname{inv}_m$.
