Does the following statement

"Let $G$ be a finitely generated group and let $X(G)$ be the $SL(2,\mathbb{C})$ character variety of $G$. If $X(G)$ contains an irreducible component $X_0$ such that for every $g\in G$ and $\chi_{\rho}\in X_0$, $\chi_{\rho}(g)$ is an algebraic number. Then $dim_{\mathbb{C}}X_0=0$."

contradict the following which is true by result of Thurston:

"The trace field of a finite covolume Kleinian group is a finite extension of $\mathbb{Q}$"

I am asking because if we take the figure-8 complement then it is a complete finite volume hyperbolic 3-manifold with finitely generated fundamental group. However the closure of the set of irreducible character has $dim_{\mathbb{C}} =1 > 0$.