Let $A(n), B(n) \in \mathbb{Z}[n]$ be polynomials, not both constant, such that $4A^3(n) + 27B^2(n)$ is not the zero polynomial and the polynomial (in variables $x, y$) $$y^2 - x^3 - A(n)x - B(n) \in \mathbb{C}(n)[x, y]$$

has no zeroes in $\mathbb{C}(n) \times \mathbb{C}(n)$. Let $K$ be a number field. Furhter, let $Z$ denote the common complex zeroes of the above polynomials when $n$ runs through $\mathbb{N} = 1, 2, 3, ...$

I wonder if it is known whether there always exists an $n_0 \in \mathbb{N}$ such that when $n = n_0$, all the zeroes of the respective polynomial that belong to $\mathcal{O}_K \times \mathcal{O}_K$ must also belong to $Z$.