Given a cubic $F(x,y,z)=x^3 + y^2 z + y z^2 = 0$ I have to find projective transformation $\phi : \mathbb{P}^2 \rightarrow \mathbb{P}^2$, such that, $\phi(F) = x^3 + axz^2 + bz^3 - y^2 z = 0$ (Weierstrass normal form).
There is an answer that touches on this subject pretty well: Irreducible cubic curve in normal form.
However doing it by the link https://ctnt-summer.math.uconn.edu/wp-content/uploads/sites/1632/2016/02/Matsuura-projective_transformation.pdf given in the answer is very tedious with some cumbersome calculations, on the other hand, ‘guessing’ the transformation seems the way to go. Now in my example, $F$ already appears to be close to normal form(by mapping $z \rightarrow -z$ I get $y^2z = x^3 + yz^2)$ so idealy would be to transform $yz^2 \rightarrow axz^2 +bz^3$ but I do not think this is possible without altering other part off the cubic.I tried something like $x \rightarrow x+z$ and $z \rightarrow x+z$ and nothing nice came out. Any ideas how to it without too much computation?