We continue our study of genus 2 curves C that admit a cover C → E to a genus 1 curve E of prime degree n. These curves C form an irreducible 2-dimensional subvariety ℒn of the moduli space ℳ2 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 ℒ5 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 ℒ5 as subvarieties of ℳ2 and classify all curves in these loci which have extra automorphisms.
© de Gruyter 2009