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We establish precise counts of cubic $\mathbb{F}_q[t]$-algebras and quartic$\mathbb{F}_q[t]$-algebras with cubic resolvent of discriminant $q^b$. To getthese counts, the main geometric input is our construction of smoothcompactifications of simply branched degree $d$ Hurwitz stacks in mixedcharacteristic for $d \in \{3,4,5\}$ as stacks of orbifold quasimaps to$B\mathbf{S}_d$, which is presented as a GIT quotient stack $[V_d^{ss}/G_d]$for a prehomogeneous vector space $(G_d,V_d)$. The main number-theoretic inputis Igusa's computations of the Igusa zeta functions for $(G_3,V_3)$ and$(G_4,V_4)$. Our compactifications of Hurwitz stacks of a curve $C$ admitproper extensions of the branch map to $\mathrm{Sym}^n(C)$. Because theseextensions are small, we can compute the intersection cohomology of certainlocal systems on $\mathrm{Conf}^n(C) \subset \mathrm{Sym}^n(C)$ as thecohomology of our compactifications, answering questions ofEllenberg-Tran-Westerland and Kapranov-Schechtman. Using a comparison theoremof Kapranov-Schechtman, we compute the part of the cohomology of a576-dimensional Nichols algebra $\mathfrak{B}_4$ that is invariant under anatural $\mathbf{S}_4$-action. Because our counts are precise, we obtain twoanswers to Venkatesh's question about the topological origin of secondary termsin arithmetic counts.

Hurwitz spaces, Nichols algebras, and Igusa zeta functions