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I would appreciate resource recommendations preferably or an outline of approaches as to how one can compute the period matrix of [itex]X[/itex], or more precisely, of the Jacobian variety of the curve.

My motivation is I am studying the Kawazumi-Zhang invariant (related to the Faltings invariant) of Riemann surfaces, which can be expressed as a Fourier series in the period matrix, and thus if I can evaluate the period matrix [itex]\Omega[/itex] of the algebraic manifold, then I can compute its Kawazumi-Zhang invariant, numerically.