Legendre parameter orbit ($\lambda ,1-\lambda ,1/\lambda ,(\lambda -1)/\lambda ,1/(1-\lambda ),\lambda /(\lambda -1)$)

For $Y^{2}Z=X(X-Z)(X-\lambda Z)$ 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 $X^{\prime }=(X-r_{0}Z)/h$, $Y^{\prime }=Y/s$, $Z^{\prime }=Z$, 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.