Simultaneous torsion in the Legendre family

We improve a result due to Masser and Zannier, who showed that the set $$\{\lambda \in \mathbb{C}\setminus \{0,1\}:(2,\sqrt{2(2-\lambda )}),(3,\sqrt{6(3-\lambda )})\in (E_{\lambda }{)}_{\text{tors}}\}$$ is finite, where $E_\lambda \colon y^2 = x(x-1)(x-\lambda)$ is the Legendre family of elliptic curves. More generally, denote by $T(\alpha ,\beta )$, for $\alpha, \beta \in {\mathbb C} \setminus {0,1}$,$\alpha \neq \beta$,thesetof$\lambda \in {\mathbb C} \setminus {0,1}$suchthatallpointswith$x$-coordinate$\alpha$or$\beta$ are torsion on $E_{\lambda }$. By further results of Masser and Zannier, all these sets are finite. We present a fairly elementary argument showing that the set $T(2,3)$ in question is actually empty. More generally, we obtain an explicit description of the set of parameters $\lambda$ such that the points with $x$-coordinate $\alpha$ and $\beta$ are simultaneously torsion, in the case that $\alpha$ and $\beta$ are algebraic numbers that not 2-adically close. We also improve another result due to Masser and Zannier dealing with the case that $\mathbb{Q}(\alpha ,\beta )$ has transcendence degree 1. In this case we show that $\mathrm{\#}T(\alpha ,\beta )\le 1$ and that we can decide whether the set is empty or not, if we know the irreducible polynomial relating $\alpha$ and $\beta$. This leads to a more precise description of $T(\alpha ,\beta )$ also in the case when both $\alpha$ and $\beta$ are algebraic. We performed extensive computations that support several conjectures, for example that there should be only finitely many pairs $(\alpha ,\beta )$ such that $#T(\alpha, \beta) \ge 3$.