Quadratic rational maps with a five-periodic critical point.

We study the moduli space $\mathrm{Per}^{\mathrm{cm}}_5(0)$ of degree-2 rational maps $\mathbb{P}^1\to\mathbb{P}^1$ that have a marked 5-periodic critical point. We show that $\mathrm{Per}^{\mathrm{cm}}_5(0)$ is an elliptic curve $\mathcal{C}_5$ punctured at 10 points, and we identify the isomorphism class of $\mathcal{C}_5$ over $\mathbb{Q}$. In order to do so, we develop techniques for using compactifications of Hurwitz spaces to study subvarieties of the moduli space of degree-$d$ rational maps defined by critical orbit relations. We carry out an experimental study of the interaction between dynamically defined points of $\mathrm{Per}^{\mathrm{cm}}_5(0)$ (such as PCF points or punctures) and the group structure of $\mathcal{C}_5$.