2-isogenies on Jacobians of genus 3 curves

We consider 2-isogenies on Jacobians of curves of genus 3. Unlike the situation in lower genera, over a non–algebraically closed base field, the codomain of such an isogeny is generally not a Jacobian of a curve, but only a quadratic twist of one. We follow the approach of Donagi–Livné and Lehavi–Ritzenthaler to determine the curve associated to the twisted Jacobian and refine the construction to additionally determine the quadratic extension of the base field that trivialises the twist in the situation where we have a flag specified on the kernel of the isogeny. We also provide Magma code of our construction, which computes both the curve associated to the twisted Jacobian and the quadratic extension required to trivialise the twist.