Let $\alpha, \beta \in {\mathbb C} \setminus \{0,1\}$ be distinct, and define $T(\alpha,\beta)$ to be the set of parameters $\lambda \in {\mathbb C} \setminus \{0,1\}$ such that the points with $x$-coordinate $\alpha$ and $\beta$ are torsion on the Legendre elliptic curve $y^2 = x(x-1)(x-\lambda)$.
Masser and Zannier have shown that $T(\alpha,\beta)$ is always finite. We will present some results regarding effectivity of $T(\alpha,\beta)$; for example, we show that the set can be effectively determined when $\alpha$ and $\beta$ are both algebraic and not too close 2-adically. We also show that $T(\alpha,\beta)$ has at most one element when ${\mathbb Q}(\alpha,\beta)$ has transcendence degree 1. Based on this result, we obtained a large amount of experimental data, and we will present some conjectures that are suggested by this.