Genus two curves covering elliptic curves: a computational approach

A genus 2 curve $C$ has an elliptic subcover if there exists a degree $n$ maximal covering $ψ: C \to E$ to an elliptic curve $E$. Degree $n$ elliptic subcovers occur in pairs $(E, E')$. The Jacobian $J_C$ of $C$ is isogenous of degree $n^2$ to the product $E \times E'$. We say that $J_C$ is $(n, n)$-split. The locus of $C$, denoted by $Ł_n$, is an algebraic subvariety of the moduli space $\M_2$. The space $Ł_2$ was studied in Shaska/Völklein and Gaudry/Schost. The space $Ł_3$ was studied in Shaska (2004) were an algebraic description was given as sublocus of $\M_2$. In this survey we give a brief description of the spaces $Ł_n$ for a general $n$ and then focus on small $n$. We describe some of the computational details which were skipped in Shaska/Völklein and Shaska (2004). Further we explicitly describe the relation between the elliptic subcovers $E$ and $E'$. We have implemented most of these relations in computer programs which check easily whether a genus 2 curve has $(2, 2)$ or $(3, 3)$ split Jacobian. In each case the elliptic subcovers can be explicitly computed.

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