Coarse moduli spaces of Weierstrass fibrations over a smooth conic curve were constructed by the classical work of [Miranda] using geometric invariant theory. In our paper, we extend this treatment by using results of [Romagny] regarding group actions on stacks to give an explicit construction of the moduli stack \({\mathcal {W}}_{n}\) of Weierstrass fibrations over a smooth conic curve with discriminant degree 12n and a section. We show that it is a smooth algebraic stack and prove that for \(n\ge 2\) , the open substack \({\mathcal {W}}_{\textrm{min},n}\) of minimal Weierstrass fibrations is a separated Deligne–Mumford stack over any base field K with \(\textrm{char}(K) \ne 2,3\) and \(\not \mid n\) . Arithmetically, for the moduli stack \({\mathcal {W}}_{\textrm{sf},n}\) of stable Weierstrass fibrations, we determine its motive in the Grothendieck ring of stacks to be \(\{{\mathcal {W}}_{\textrm{sf},n}\} = {\mathbb {L}}^{10n - 2}\) in the case that n is odd, which results in its weighted point count to be \(\#_q({\mathcal {W}}_{\textrm{sf},n}) = q^{10n - 2}\) over \({\mathbb {F}}_q\) . In the appendix, we show how our methods can be applied similarly to the classical work of [Silverman] on coarse moduli spaces of self-maps of the projective line, allowing us to construct the natural moduli stack and to compute its motive.
