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We extend the explicit quadratic Chabauty methods developed in previous workby the first two authors to the case of non-hyperelliptic curves. This resultsin 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 numbergreater than one, and which satisfies some additional conditions. Thisalgorithm is then applied to the modular curve $X_{s}(13)$, completing theclassification of non-CM elliptic curves over $\mathbf{Q}$ with split Cartanlevel structure due to Bilu-Parent and Bilu-Parent-Rebolledo.

Explicit Chabauty-Kim for the Split Cartan Modular Curve of Level 13