Constructing genus 2 curves with given refined Humbert invariants

In 1994, Kani introduced an algebraic version of the Humbert invariant, known as the refined Humbert invariant. This invariant q_C is a positive definite quadratic form attached to a smooth curve C of genus 2. It serves as a vital tool, as many geometric properties of C are reflected in the arithmetic properties of q_C. When the Jacobian J_C of a genus 2 curve C is isogenous to a product of an elliptic curve with complex multiplication, the forms q_C have been completely classified recently. In this paper, building upon this classification, we present a constructive algorithm that produces J_C and a divisorial representative of a curve C of genus 2 such that its refined Humbert invariant q_C is equivalent to a given integral ternary quadratic form.