Binary dihedral invariant hypersurface ($W^2-V{U}^2+4V^{n+1}=0$)

Let $\zeta =e^{\pi i/n}$ and let the binary dihedral group act on ${\mathbb{C}}^2$ by $(x,y)\mapsto (\zeta x,{\zeta }^{-1}y)$ and $(x,y)\mapsto (-y,x)$. For $n\ge 2$, the invariantspresent the invariant ring as $\mathbb{C}[U,V,W]/(W^2-V{U}^2+4V^{n+1})$. First take the cyclic invariants $A=x^{2n},B=y^{2n},C=xy$ with $AB=C^{2n}$. The residual involution interchanges $A,B$ and negates $C$, so invariant normal forms are polynomials in $A+B,C^2$ plus $C(A-B)$ times such polynomials. For $n=3$ the group has order twelve and the equation is $W^2-V{U}^2+4V^4=0$.