Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves

This paper provides empirical evidence for the Birch and Swinnerton-Dyer co njectures for modular Jacobians of genus 2 curves. The second of these conj ectures relates six quantities associated to a Jacobian over the rational n umbers. One of these six quantities is the size of the Shafarevich-Tate gro up. Unable to compute that, we computed the five other quantities and solve d fur the last one. In all 32 cases, the result is very close to an integer that is a power of 2. In addition, this power of 2 agrees with the size of the 2-torsion of the Shafarevich-Tate group, which we could compute.