An arithmetic invariant theory of curves from $E_8$

Let $k$ be a field of characteristic 0, let $C/k$ be a uniquely trigonal genus 4 curve, and let $P \in C(k)$ be a simply ramified point of the uniquely trigonal morphism. We construct an assignment of an orbit of an algebraic group of type $E_8$ acting on a specific variety to each element of $J_C(k)/2$. The algebraic group and variety are independent of the choice of $(C,P)$. We also construct a similar identification for uniquely trigonal genus 4 curves $C$ with $P \in C(k)$ a totally ramified point of the trigonal morphism. Our assignments are analogous to the assignment of a genus 3 curve with a rational point $(C,P)$ to an orbit of an algebraic group of type $E_7$ exhibited by Jack Thorne. Our assignment is also analogous to one constructed by Bhargava and Gross, who use it determine average ranks of hyperelliptic Jacobians.