Two-isogeny halving criterion for the alpha family ($(1,1)\in 2C(\mathbb{Q})⟺\alpha \in {\mathbb{Z}}^2\text{ or }\alpha +4\in {\mathbb{Z}}^2$)

For $C:y^2=x^3+\alpha x^2-\alpha x$, $\alpha \in \mathbb{Z}\setminus \{0,-4\}$, the two-isogeny formula companion is $D:v^2=u(u^2-2\alpha u+\alpha (\alpha +4))$. With the positive-sign standard isogenies, the two preimages of $(1,1)$ under the dual are $(\alpha ,-2\alpha )$ and $(\alpha +4,2(\alpha +4))$. The two-torsion square-class homomorphism on $D$ detects membership in the image of the forward isogeny of elliptic curves by the class of $u$. Thus either preimage lifts precisely when its first coordinate is a rational square. Their being integers turns rational squareness into integer squareness.