To get some insight on the zero locus of the cubic curve, I've tried a couple of online visualisers on Google and mostly failed to generate the plot (run time errors...) except Wolfram Alpha which gave me the following;
Is this visualisation correct??
Maybe first divide by $z^3$ in order to get inhomogeneous coordinates? This yields
$$ C':y^2-x^3-x^2-x-1=0 \quad(z\neq0) $$ (which is effectively using $z=1$). And for $z=0$ if follows $x=0$ and hence $y\neq 0$, sometimes called "point at infinity".
With the dimension down to 2 it's easier to visualize, like in this Desmos plot.