Genus 2 curves that admit a degree 5 map to an elliptic curve

TL;DR

This paper investigates genus 2 curves with degree 5 maps to elliptic curves, providing explicit normal forms, equations, parametrizations, and automorphism classifications for these special curves.

Contribution

It extends previous work on degrees 2 and 3 to degree 5, offering explicit descriptions and classifications of genus 2 curves with degree 5 elliptic subcovers.

Findings

Computed normal forms for genus 2 curves in the degree 5 locus.

Derived equations of elliptic subcovers for these curves.

Classified curves with extra automorphisms within the locus.

Abstract

We continue our study of genus 2 curves $C$ that admit a cover $ C \to E$ to a genus 1 curve $E$ of prime degree $n$. These curves $C$ form an irreducible 2-dimensional subvariety $\L_n$ of the moduli space $\M_2$ of genus 2 curves. Here we study the case $n=5$. This extends earlier work for degree 2 and 3, aimed at illuminating the theory for general $n$. We compute a normal form for the curves in the locus $\L_5$ and its three distinguished subloci. Further, we compute the equation of the elliptic subcover in all cases, give a birational parametrization of the subloci of $\L_5$ as subvarieties of $\M_2$ and classify all curves in these loci which have extra automorphisms.