Selmer's curve $3X^3+4y^3+5z^3=0$ is a non-example (see the comment below) but somwhat relevant. See Theorem 1 in Mazur's article titled [ON THE PASSAGE FROM LOCAL TO GLOBAL IN NUMBER THEORY][1].

  [1]: http://www.ams.org/bull/1993-29-01/S0273-0979-1993-00414-2/S0273-0979-1993-00414-2.pdf