We consider isogenies on Jacobians J of genus 3 curves with a
kernel that is a maximal isotropic subgroup of the 2-torsion J[2] and
confront a phenomenon that is new in genus 3: for genus 1 and 2 the
codomain is generally again a Jacobian of a curve and we have
an explicit construction of that curve. In the genus 3 case we only
obtain that the codomain is a quadratic twist of a Jacobian.

We use a construction by Donagi-Livne, refined by Lehavi-Ritzenthaler
that constructs the curve whose Jacobian is the codomain up to quadratic
twist. We refine the construction further to explicitly determine this
quadratic twist and use it to compute many examples. The construction
requires the specification of a flag on the isogeny kernel and
constructs the codomain in steps, in terms of various Prym varieties.