$2y^2=x^4-17$ ($1951$, Lind) is a well-known counterexample that violates the local-global principle.
This is an order two elements of $Sha(E:y^2=x^3+17x/\Bbb{Q})$, and it has been proven that the order of $Sha(E:y^2=x^3+17x/\Bbb{Q})$ is exactly $4$.
But what are the equations of the other $2$ non-trivial elements of $Sha(E:y^2=x^3+17x/\Bbb{Q})$ ?
We should look for a curve that has a local solution but no global solution, and where $E:y^2=x^3+17x$ acts simply transitively. Such curves should be of the form $qy^2=x^4-p$ (where $p$ and $q$ are integers), but I cannot figure them(2 curves) out.
The Arithmetic of Elliptic Curves. In the proof of Proposition 6.5(a), it notes that for $p\equiv1\pmod8$, the three non-trivial elements of the $2$-Selmer group of $y^2=x^3+px$ are $C_{-1}$, $C_{2}$ and $C_{-2}$, where $$ C_d : dw^2 = d^2 - 4pz^4. $$ Let $E$ be the curve $y^2=x^3+17x$. Then up to change of variables, the three non-trivial elements of $I\!I\!I(E)$ are represented by the homogeneous spaces $$ C_{-1}:-w^2=-68z^4+1, \qquad C_2 : 2w^2 = -68z^4+4,\qquad C_{-2}: -2w^2 = 68z^4+4. $$