The following is probably well-known (I'd appreciate a link):

for a field $K$ that is a finite extension of the field of rational numbers, give a polynomial $f(x,y) ∈ Q[x,y]$ of the form $y^2 − x^3 − Ax − B$, s.t. $4A^3 + 27B^2 ≠ 0$ and $f(x,y)$ has no zero in $K^2$.

Thank you - Albertas

with a distinguished point over $K$. $\endgroup$5more comments