Given a generic degree-2 cover of a genus 1 curve D by a non hyperelliptic
genus 3 curve C over a field k of characteristic different from 2, we produce
an explicit genus 2 curve X such that Jac(C) is isogenous to the product of
Jac(D) and Jac(X). This construction can be seen as a degenerate case of a
result by Nils Bruin.
Hanselman, Jeroen; Schiavone, Sam; Sijsling, Jeroen, 2021, Gluing Curves Of Genus 1 And 2 Along Their 2-Torsion, Mathematics Of Computation, 90, 331, pp. 2333-2379, 10.1090/mcom/3627.