  We extend the explicit quadratic Chabauty methods developed in previous work
by the first two authors to the case of non-hyperelliptic curves. This results
in an algorithm to compute the rational points on a curve of genus $g \ge 2$
over the rationals whose Jacobian has Mordell-Weil rank $g$ and Picard number
greater than one, and which satisfies some additional conditions. This
algorithm is then applied to the modular curve $X_{s}(13)$, completing the
classification of non-CM elliptic curves over $\mathbf{Q}$ with split Cartan
level structure due to Bilu-Parent and Bilu-Parent-Rebolledo.
