Trinomials Defining Quintic Number Fields

Given a quintic number field K/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 4 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 CK using elliptic curve Chabauty.