  In a recent work of Duke, Imamo\={g}lu, and T\'{o}th, the linking number of
certain links on the space
$\text{SL}(2,\mathbb{Z})\backslash\text{SL}(2,\mathbb{R})$ is investigated.
This linking number has an alternative interpretation as the intersection
number of closed geodesics on the modular curve, which is the focus of this
paper. By relating the intersection number to a combinatorial computation
involving rivers of Conway topographs, an efficient algorithm for computing
intersection numbers is produced. A formula for the total intersection of a
pair of positive coprime fundamental discriminants is also derived, which can
be thought of as a real quadratic analogue of a classical result of Gross and
Zagier. The paper ends with numerical computations and distribution questions
relating to intersection numbers.
