Determining all \((2, 3)\)-torus structures of a symmetric plane curve (Q1721827)

Let \(f \in \mathbb C[u,v,w]\) be a reduced non-constant homogeneous polynomial and \(C \subset \mathbb P^2\) the corresponding plane curve. A \((p,q)\)-torus structure of \(C\) consists of two homogeneous polynomials \(g,h \in \mathbb C[u,v,w]\) such that \(g^p+h^q=f\). The 39-cuspidal degree 12 curve \(C\) defined by \[ f=-v^{12}-w^{12}-3v^6w^6-2v^3w^9-2v^9w^3 +(-12v^3w^6-12v^6w^3+12w^9+12v^9)u^3 \] \[ +(-42v^6-42w^6+138v^3w^3)u^6+(36w^3+36v^3)u^9+27u^{12} \] is studied and all \((2,3)\)-torus structures of this curve are described.