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We show that, up to a natural equivalence relation, the only non-trivial,non-identity holomorphic maps$\mathrm{Conf}_n\mathbb{C}\to\mathrm{Conf}_m\mathbb{C}$ between unorderedconfiguration spaces, where $m\in\{3,4\}$, are the resolving quartic map$R\colon\mathrm{Conf}_4\mathbb{C}\to\mathrm{Conf}_3\mathbb{C}$, a map$\Psi_3\colon\mathrm{Conf}_3\mathbb{C}\to\mathrm{Conf}_4\mathbb{C}$ constructedfrom the inflection points of elliptic curves in a family, and $\Psi_3\circ R$.This completes the classification of holomorphic maps$\mathrm{Conf}_n\mathbb{C}\to\mathrm{Conf}_m\mathbb{C}$ for $m\leq n$,extending results of Lin, Chen and Salter, and partially resolves a conjectureof Farb. We also classify the holomorphic families of elliptic curves over$\mathrm{Conf}_n\mathbb{C}$. To do this we classify homomorphisms between braidgroups with few strands and $\mathrm{PSL}_2\mathbb{Z}$, then apply powerfulresults from complex analysis and Teichm\"uller theory. Furthermore, we prove aconjecture of Castel about the equivalence classes of endomorphisms of thebraid group with three strands.

Braid groups, elliptic curves, and resolving the quartic