Plane model y plus x cubed plus xy cubed equals zero of the Klein quartic ($y+x^3+x{y}^3=0$)

The projective completion ofis the smooth plane quarticThe coordinate relabelling $[A:B:C]=[Y:X:Z]$ turns this equation into the standard Klein quartic equation $A^{3}B+B^{3}C+C^{3}A=0$.