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