Counting $D_4$-field extensions by multi-invariants

We count the number of Galois extensions $M\mathrm{/}\mathbb{Q}$ with fixed Galois group $\text{Gal}(M\mathrm{/}\mathbb{Q})=D_4$ ordered by multi-invariants introduced by Gundlach. We verify the asymptotic behavior predicted by Gundlach’s version of Malle’s conjecture. We compare the leading constant to recent predictions by Loughran and Santens.