Genus-rank difference (awaiting review)

The genus-rank difference of a modular curve $X_H$ of positive genus is the integer obtained by subtracting the analytic rank of $X_H$ from its genus.

This difference is of interest because the methods available to compute the set of rational points on $X_H$ depend critically on whether the Mordell-Weil rank of its Jacobian is less than, equal to, or greater than its genus (one expects the Mordell-Weil rank to equal the analytic rank of $X_H$ and the later is much more feasible to compute).