Explicit computation of the field of moduli of some non-hyperelliptic pseudo-real curves

For each even integer \(k \ge 2\), we construct an explicit real two-dimensional family \(C^{(k)}_{r,\theta }\) 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.