Gluing Genus 1 and Genus 2 Curves Along $\mathrm{\ell }$-torsion

Let $Y$ be a genus $2$ curve over $\mathbb{Q}$. We provide a method to systematically search for possible candidates of a prime $\mathrm{\ell }\ge 3$ and a genus $1$ curve $X$ for which there exists a genus $3$ curve $Z$ over $\mathbb{Q}$ whose Jacobian is, up to quadratic twist, $(\mathrm{\ell },\mathrm{\ell },\mathrm{\ell })$-isogenous to the product of Jacobians of $X$ and $Y$, building on the work by Hanselman, Schiavone, and Sijsling for $\mathrm{\ell }=2$. We find several such pairs $(X,Y)$ for prime $\mathrm{\ell }$ up to $13$. We also improve their numerical gluing algorithm, allowing us to successfully glue genus $1$ and genus $2$ curves along their $13$-torsion.