Let $X = \text{Spec }B$ be a smooth affine curve over $\mathbb{C}$ with finitely many automorphisms. Ie, $B$ is a smooth $\mathbb{C}$-algebra with finitely many $\mathbb{C}$-algebra automorphisms.

Let $A\subset B$ be a $\overline{\mathbb{Q}}$-subalgebra such that the inclusion $A\hookrightarrow B$ induces an isomorphism $A\otimes_{\overline{\mathbb{Q}}}\mathbb{C}\cong B$. Ie, $A$ generates $B$ as a $\mathbb{C}$-algebra.

Given that such an $A$ exists, is it *unique*? (I don't mean unique up to isomorphisms - I literally mean that it is the **only** $\overline{\mathbb{Q}}$-subalgebra of $B$ satisfying the above properties.)

Note that for $X = \text{Spec }\mathbb{C}[t]$, the statement is false - there are infinitely many $\overline{\mathbb{Q}}$-subalgebras which generate $\mathbb{C}[t]$. For example, you can take $\overline{\mathbb{Q}}[t+n\pi]$ for $n\in\mathbb{Z}$. However, this doesn't satisfy the hypotheses since $\mathbb{C}[t]$ has the whole affine group as automorphisms.