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Bring's Curve: its Period Matrix and the Vector of Riemann Constants

Bring's curve is the genus 4 Riemann surface with automorphism group of maximal size, $S_5$. Riera and Rodríguez have provided the most detailed study of the curve thus far via a hyperbolic model. We will recover and extend their results via an algebraic model based on a sextic curve given by both Hulek and Craig and implicit in work of Ramanujan. In particular we recover their period matrix; further, the vector of Riemann constants will be identified.