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 $Z$.



One parameter families of elliptic curves over rings of integers of number fields
Let $E\_n$ denote your elliptic curve. It's probably easier to ask for an integer $n\_0$ such that the Mordell-Weil groups $E\_{n\_0}(\mathbb{Q})$ and $E\_{n\_0}(K)$ coincide. There has been a fair amount of attention given to the question of elliptic curves $E/K$ and extensions $L/K$ such that both $E(K)$ and $E(L)$ have rank 1, because this turns out to be useful in studying Hilbert's 10th problem. So the following article (and its reference list) might be helpful for your problem:


MR2041072:
Bjorn Poonen, Using elliptic curves of rank one towards the undecidability of Hilbert's tenth problem over rings of algebraic integers. *Algorithmic number theory* (Sydney, 2002), 33–42, *Lecture Notes in Comput. Sci*., **2369**, Springer, Berlin, 2002


