= Legendre parameter orbit
{c}
{title2=$\lambda,\ 1-\lambda,\ 1/\lambda,\ (\lambda-1)/\lambda,\ 1/(1-\lambda),\ \lambda/(\lambda-1)$}
For $Y^2Z=X(X-Z)(X-\lambda Z)$ over $\mathbb C$, order its three finite roots as $(r_0,r_1,r_2)$ and put $h=r_1-r_0$. The <projective transformation> $X'=(X-r_0Z)/h$, $Y'=Y/s$, $Z'=Z$, with $s^2=h^3$, gives <Legendre form of an elliptic curve> with parameter $(r_2-r_0)/(r_1-r_0)$. The six root orderings give the six displayed values, all different from zero and one. These exhaust changes that act affinely on $X/Z$ and scale $Y/Z$, because such a change must send the finite root set onto the new finite root set. There are six distinct values except that $\lambda\in\{-1,2,1/2\}$ gives three and $\lambda^2-\lambda+1=0$ gives two.
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