= Two-isogeny halving criterion for the alpha family
{title2=$(1,1)\in2C(\mathbb Q)\iff\alpha\in\mathbb Z^2\text{ or }\alpha+4\in\mathbb Z^2$}
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.
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