Arithmetic of elliptic curves induced by regular Diophantine triples

We study elliptic curves induced by regular Diophantine triples, with emphasis on their torsion subgroups. We show that an elliptic curve $E$ induced by a regular Diophantine triple in integers necessarily has torsion subgroup $E(\mathbb{Q})_{\mathrm{tors}} \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$. Moreover, we develop a criterion for when such an elliptic curve acquires a point of order $3$ over a quadratic field. For a particular family $\{ k-1, k+1, 4k\}$, we use it to show that this does not happen. Finally, we study both the torsion and the generic rank of a family of elliptic curves induced by the $D(-k^2)$-triple $\{1, 2k^2, 2k^2+2k+1\}$.