Solution
= Solution
Compose the preceding transformations, applying $\sigma_\lambda$ first and then $\tau_{1/\lambda}$. With the square-root convention already specified,
$$
\boxed{\upsilon_\lambda[x:y:z]
=[z-x/\lambda:iy/(\lambda s):z],\qquad s^2=\lambda.}
$$
This invertible <projective transformation> fixes $[0:1:0]$ and carries $C_\lambda$ to $C_{1-1/\lambda}$. It is important to retain the sign of the chosen lift on the $y$ coordinate when computing its order later.