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We give an algebraic method to compute the fourth power of the quotient ofany even theta constants associated to a given non-hyperelliptic curve in termsof geometry of the curve. In order to apply the method, we work outnon-hyperelliptic curves of genus 4, in particular, such curves lying on asingular quadric, which arise from del Pezzo surfaces of degree 1. Indeed, weobtain a complete 2-level structure of the curves by studying their thetacharacteristic divisors via exceptional divisors of the del Pezzo surfaces asthe structure is required for the method.

A Thomae-like Formula: Algebraic Computations of Theta Constants