 Legendre form of an elliptic curve

Over an algebraically closed field of characteristic different from two, a cubic Weierstrass model with distinct roots becomes this form by sending two roots to zero and one, then scaling $Y$ by a square root of the resulting nonzero coefficient. The third root becomes $\lambda \notin \{0,1\}$. The point $[0:1:0]$ is a flex and the curve is a genus one curve.