Genus two curves covering elliptic curves: a computational approach

T. Shaska
2005 Computational Aspects of Algebraic Curves  
A genus 2 curve C has an elliptic subcover if there exists a degree n maximal covering ψ : C → 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 × E . We say that J C is (n, n)-split. The locus of C, denoted by Ln, is an algebraic subvariety of the moduli space M 2 . The space L 2 was studied in Shaska/Völklein [32] and Gaudry/Schost [10]. The space L 3 was studied in [26] were an algebraic description
more » ... was given as sublocus of M 2 . In this survey we give a brief description of the spaces Ln for a general n and then focus on small n. We describe some of the computational details which were skipped in [32] and [26] . 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.
doi:10.1142/9789812701640_0013 fatcat:rnilovvsv5effext2ztkmu7q3i