On the Parametrization of Hyperelliptic Fields with $S$-Units of Degrees 7 and 9

We show that if $k$ is an algebraically closed field with $\operatorname{char}k=0$, then the set of polynomials $f$ of degree $5$ such that the field $k(x)(\sqrt{f}\,)$ has a nontrivial $S$-unit of degree $7$ or $9$ and the continued fraction expansion of $\sqrt{f}/x$ is periodic is a one-parameter set corresponding to a rational curve with finitely many deleted points.