On the Mordell-Weil rank and $2$-Selmer group of a family of elliptic curves

We consider the parametric family of elliptic curves over $\mathbb{Q}$ of the form $E_m:y^2=x(x-n_1)(x-n_2)+t^2$, where $n_1$, $n_2$ and $t$ are particular polynomial expressions in an integral variable $m$. In this paper, we investigate the torsion group $E_m(\mathbb{Q}{)}_{\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}}$, a lower bound for the Mordell-Weil rank $r(E_m)$ and the $2$-Selmer group ${\mathrm{S}\mathrm{e}\mathrm{l}}_2(E_m)$ under certain conditions on $m$. This extends the previous works done in this direction, which are mostly concerned with the Mordell-Weil ranks of various parametric families of elliptic curves.