In this talk we shall expose a concrete relation between the algebraic and transcendental parameters of a nonsingular $z_n$ curve.A nonsingular $z_n$ curve is a compact Riemann surface with algebraic equation $$ w^n=\prod^{j=nr−2}_{j=0}(z − \lambda_j)$$with $\lambda_i \neq\lambda_j$ for $i \neq j$, and $r\ge 2$. Thus the Riemann surface is represented as an $n$ sheeted cover of the sphere branched over the $nr-1$ points $\lambda_0, ..., \lambda_{nr−2}$ and the point at $\infty$. We shall also here assume that $\lambda_0 = 0$, $\lambda_1 = 1$. We shall express the quantities $\lambda_i$ in terms of theta constants with rational characteristics of order $n$ and these will give rise to theta constant identities.