How can I find the inverse of the map $\phi \colon E_1 \to E_2$ with $(x,y) \mapsto \left(\frac{y^2}{x^2},\frac{y(2-x^2)}{x^2}\right)$ where $E_1 : y^2 = x^3+3x^2+2x$ and $E_2 : y^2 = x^3-6x^2+x$?
Also, are the infinity points $\mathcal{O}$ and $\mathcal{O'}$, respectively, for both curves the same? $(0:1:0)$ (this is in preojective coordinate, don't know how to write in affine coordinates), I need $\phi^{-1}(\mathcal{O'})$.