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

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 ${Ł}_n$ 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 ${Ł}_5$ 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 ${Ł}_5$ as subvarieties of $\M_2$ and classify all curves in these loci which have extra automorphisms.

Cited by in corpus (1)

• The arithmetic of genus two curves with (4,4)-split Jacobians