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Consider the genus $43$ Riemann surface $X$ obtained from the desingularization and compactification of the plane algebraic curve

$C: f(x,y) = y^3 \big((y^4 - 16)^2 - 2x\big) - x^{12}.$

The Riemann matrix (normalized period matrix) of this curve is given by:

(Click to expand.) This matrix is indeed symmetric and its imaginary part is positive definite.