Bring's curve: Old and New

TL;DR

This paper reviews and extends the properties of Bring's curve, a genus-4 Riemann surface with automorphism group S_5, using modern computational tools to provide new proofs, classifications, and connections to modular curves.

Contribution

It offers new proofs, complete automorphism group realization, and identifies novel elliptic quotients and invariants of Bring's curve, enhancing understanding of its structure.

Findings

Complete realization of automorphism group for a plane curve model

Identification of a new elliptic quotient related to modular curve X_0(50)

Description of the orbit decomposition of theta characteristics

Abstract

Bring's curve, the unique Riemann surface of genus-4 with automorphism group $S_5$, 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 $X_0(50)$, providing a complete description of the orbit decomposition of the theta characteristics, and identifying the unique invariant characteristic with the divisor of the Sz\"ego kernel. In achieving this we have used modern computational tools in Sagemath, Macaulay2, and Maple, for which notebooks demonstrating calculations are provided.