$(G,F)$-points on Q-algebraic varieties

Let G∈ Q[x,y,z] be a polynomial, and let $V(G)$ be the Q-algebraic variety corresponding to G, i.e., V(G)=³~|~G(P)=0\. Let \[{split} F: &Q^3→ Q^3,\\ &(x,y,z)↦ (f(x),f(y),f(z)) {split}\] be a vector function, where f∈ Q[x]. It is easy to know that the function obtained by the composition of G and F, denoted as G∘ F, is still in Q[x,y,z]. Moreover, let V(G∘ F) be the Q-algebraic variety corresponding to G∘ F, i.e., V(G∘ F)=³~|~G∘ F(P)=0\. A rational point P is called a $(G,F)$-point on $V(G)$ if P belongs to the intersection of $V(G)$ and V(G∘ F), that is P∈ V(G)∩ V(G∘ F). Denote G,F as the set consisting of all $(G,F)$-points on $V(G)$. Obviously, G,F is a Q-algebraic variety. In this paper, we consider the algebraic variety G,F for some specific functions G and F. For these specific functions G and F, we prove that G,F will be isomorphic to a certain elliptic curve. We also analyze some properties of these elliptic curves.