= Two-torsion square-class homomorphism
{title2=$\alpha_T(P)=[x(P)-x(T)]$}
Let $E:y^2=f(x)$ with $f$ monic and separable, and let $T=(\theta,0)$. Over $K$ containing $\theta$, the map $\alpha_T(P)=[x(P)-\theta]$ for $P\ne O,T$, completed by $\alpha_T(O)=1$ and $\alpha_T(T)=[f^{\prime}(\theta)]$, is a <group homomorphism> to the <square-class group of a field>. A nonvertical chord has the product of its three $x-\theta$ values a square; a chord through $T$ instead has the product of the other two values equal to $f^{\prime}(\theta)$. These identities include tangencies.
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