Counting isomorphism classes of elliptic curves over ${\mathbb{F}}_q(t)$

We determine the precise number of isomorphism classes of elliptic curves over ${\mathbb{F}}_q(t)$ with $\text{char}({\mathbb{F}}_q)=3,2$. The key idea is to obtain the exact unweighted number of rational points on the classifying stacks $\mathcal{B}{Q}_{12}$, $\mathcal{B}{Q}_{24}$ and $\mathcal{B}Z$, where $Q_{12}$ and $Q_{24}$ denote the dicyclic groups of orders 12 and 24, respectively, and $Z$ denotes the non-reduced group scheme of order 2. This computation, inspired by the classical work of [de Jong] and performed via motivic height zeta functions of height moduli spaces constructed in [Bejleri-Park-Satriano], establishes a complete determination of the total number of isomorphism classes of rational points on ${\bar{\mathcal{M}}}_{1,1}$ over any rational function field $k(t)$ with perfect residue field $\text{char}(k)\ge 0$.