Computing the Field of moduli of some non-hyperelliptic pseudo-real curves

The explicit computation of the field of moduli of a closed Riemann surface is, in general, a difficult task. In this paper, for each even integer $k \geq 2$, we consider a suitable $2$-real parameter family of non-hyperelliptic pseudo-real Riemann surfaces of genus $g=1+(2k-3)k^{4}$. For each of them, we compute its field of moduli and also a minimal field of definition.