The Constant in Thomae-Type Formulas for Eight Points on the Complex Projective Line

We consider the family of cyclic fourfold covers $w^4 = \prod_{j=1}^{7}(z-x_j)$ of the complex projective line branched at the eight points $x_1,\ldots,x_7,\infty$. The period map identifies the configuration space $X(2,8)$ of the branch points with a Zariski open subset of a quotient of the five-di