I will explain two ways in which the abstract notion of the Jacobian of a curve can be made concrete enough for a computer to deal with. These two descriptions of the Jacobian both have their own strengths and weaknesses. I will explain how one (and the computer) can translate back and forth between the two descriptions. This allows one to have the best of both worlds.

These methods can be used to prove that two genus 2 curves are isogenous or that a genus 2 curve has complex multiplication.