An Extension to the Gusi\'c-Tadi\'c Specialization Criterion

Let $E/\mathbb Q(t)$ be an elliptic curve and let $t_0 \in \mathbb Q$ be a rational number for which the specialization $E_{t_0}$ is an elliptic curve. In 2015, Gusi\'c and Tadi\'c gave an easy-to-check criterion, based only on a Weierstrass equation for $E/\mathbb Q(t)$, that is sufficient to conclude that the specialization map at $t_0$ is injective. The criterion critically requires that $E$ has nontrivial $\mathbb Q(t)$-rational 2-torsion points. In this article, we explain how the criterion can be used in some cases where this requirement is not satisfied and provide some examples.