= Solution
A singular point would satisfy $p=0$, $p_w=2w=0$ and $p_z=-3z^2+2z+1=0$. Thus $w=0$ and $z=1$ or $-1/3$. But $p(1,0)=1$ and $p(-1/3,0)=-5/27$, both nonzero. \b[The curve is smooth everywhere.]
The complex <implicit function theorem> says that near a zero of a holomorphic function with a nonzero partial derivative, the corresponding variable is a uniquely defined holomorphic function of the others. Where $p_w\ne0$, use $z$ as a local coordinate with $w=w(z)$; where $p_z\ne0$, use $w$ with $z=z(w)$. On overlaps the coordinate transitions are holomorphic with nonzero derivatives. These charts give $Z$ a <Riemann surface> structure, and both coordinate projections are <holomorphic> in them.
A <ramification point> of a nonconstant <holomorphic map> is a point where in local coordinates the map has the form $u\mapsto u^e$ with $e\geq2$; its ramification, or branching, order is $e-1$. For $g=w$, take $(z,w)=(1,i)$. Here $p_w=2i\ne0$, so $z$ is a local coordinate. Differentiating $w(z)^2=z^3-z^2-z$ gives $w'(1)=0$ and $2i w''(1)=4$. Therefore
$$
w(z)-i=\frac1i(z-1)^2+O((z-1)^3).
$$
The nonzero quadratic coefficient gives \b[local degree $\boxed2$ and branching order $\boxed1$]. If “branching order” denotes local degree rather than its excess in a different convention, the corresponding value is two.
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