I'm working on a HW problem from one of MITs open course (Intro to Arith Geom). The problem is to show that the cubic curve $x^3+y^3+z^3=0$ can be transformed, with a suitable change of variables, to the form $y^2z=x^3+Axz^2+Bz^3$, where the particular transformation maps the point P=(1:-1:0) -> (0:1:0). (constants are over a field K with characteristic not 2 or 3)
I've tried to approach this just trying various transformations in an ad hoc manner, but kept getting stuck with unwanted cross terms. So instead I assumed each variable transforms as some linear combo of the others, i.e., $x \rightarrow ax+by+cz$. Then I plug these into the diagonal form above and get the terms I want ($y^2, xz^2,$ etc.) plus many terms I don't ($y^3, x^2z,$ etc.). This leaves me with 6 equations for the constants in front of the unwanted terms that I want to vanish + 3 equations from the requirement that P transforms as above. So 9 equations with 9 unknowns. Feels like that's a good sign?
My questions are: (1) am I on the right track here? (is this a method that will produce the right answer should I be willing to see through the algebra?) (2) is there simpler way to approach this problem that I'm not seeing?
The diagonal form looks similar to something called the Hessian form of an elliptic curve? But everything I can find online about transforming to/from Hessian form uses some additional machinery - "flexes"?