Trinomials defining quintic number fields

Given a quintic number field $K\mathrm{/}\mathbb{Q}$, we study the set of irreducible trinomials, polynomials of the form $x^5+ax+b$, that have a root in $K$. We show that there is a genus four curve $C_K$ whose rational points are in bijection with such trinomials. This curve $C_K$ maps to an elliptic curve defined over a number field, and using this map, we are able (in some cases) to determine all the rational points on $C_K$ using elliptic curve Chabauty.