  We describe a general algorithm for computing intersection pairings on
arithmetic surfaces. We have implemented our algorithm for curves over $\mathbb
Q$, and we show how to use it to compute regulators for a number of Jacobians
of smooth plane quartics, and to numerically verify the conjecture of Birch and
Swinnerton-Dyer for the Jacobian of the split Cartan curve of level 13, up to
squares.
