This talk considers the problem of producing pairs consisting of a genus 1 curve X and a genus 2 curve Y that are gluable, i.e., such that there exists a genus 3 curve Z whose Jacobian is (ℓ, ℓ, ℓ)-isogenous to Jac(X) × Jac(Y), where ℓ is a prime number. We discuss two results. First, for a fixed genus 2 curve Y, we provide a systematic way to search for a prime number ℓ and a genus 1 curve X that could produce such a genus 3 curve Z. Next, we improve a numerical algorithm to construct such a curve Z, allowing us to compute Z for ℓ up to 13. This is based on a joint work with Noah Walsh.