Malle's number field counting conjecture

Let $k$ be a number field and $G$ a transitive permutation group of degree $n$. Write $\mathcal{F}_{\disc,k}(G;X)$ for the set of $G$-extensions $L/k$ with $|\disc(L/k)|\leq X$, where the absolute value denotes the norm to $\mathbb{Q}$. Malle's number field counting conjecture. There exist positive constants $a,b,c>0$, depending on $k$ and $G$, such that

$$\#\mathcal{F}_{\disc,k}(G;X)\sim cX^{1/a}(\log X)^{b-1}$$

as $X\to\infty$. This is the general asymptotic form proposed by Malle, building on results for several specific groups and related counting problems. The conjecture also predicts the exponent constant $a$ in terms of the permutation action of $G$, as recorded separately below.