I would like to find "simple" complex algebraic curves (i.e. low genus) on a complex abelian surface $A$ (which are not just abelian subvarieties or translates of them). For example, a genus 2 curve, if such a thing existed, would be nice. The more explicitly such a curve can be described, the better.

The only way I know how to produce curves in $A$ is the general method of embedding it into a projective space and taking sections with hyperplanes. I expect this will probably give curves of quite large genus. Are there more specialized ways to produce curves in the setting of abelian surfaces? Are there nice examples of known simple curves?