Genus 3 curves whose Jacobians have endomorphisms by Q(ζ7+ζ¯7)

In this work we consider constructions of genus 3 curves X   such that End(Jac(X))⊗QEnd(Jac(X))⊗Q contains the totally real cubic number field Q(ζ7+ζ¯7). We construct explicit two-dimensional families defined over Q(s,t)Q(s,t) whose generic member is a nonhyperelliptic genus 3 curve with this property. The case when X is hyperelliptic was studied in Hoffman and Wang (2013). We calculate the zeta function of one of these curves. Conjecturally this zeta function is described by a modular form.