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 $x_1,\dots ,x_7,\mathrm{\infty }$. 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-dimensional complex ball. The inverse of the period map is expressed projectively by $105$ automorphic forms $f_J$, which are proportional to the signed branch-point polynomials ${\widehat{x}}_J$ with a common scalar factor. We determine this factor for the period $\eta$ of the differential $dz\mathrm{/}w$: it is the product of the constant $-1\mathrm{/}(2^{12}\mathrm{\Gamma }(3\mathrm{/}4{)}^{16})$ and the square of a quadratic form in $\eta$.