Bring’s curve: old and new

Abstract

Bring’s curve, the unique Riemann surface of genus 4 with automorphism group S5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_5$$\\end{document}, has many exceptional properties. We review, give new proofs of, and extend a number of these including giving the complete realisation of the automorphism group for a plane curve model, identifying a new elliptic quotient of the curve and the modular curve X0(50)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X_0(50)$$\\end{document}, providing a complete description of the orbit decomposition of the theta characteristics, and identifying the unique invariant characteristic with the divisor of the Szegő kernel. In achieving this we have used modern computational tools in Sagemath, Macaulay2, and Maple, for which notebooks demonstrating calculations are provided.