Does $2y^2=x^4-17$ have solution in $\Bbb{Q}(\sqrt{-3})$ ?
It is well known that the equation has no solution in $\Bbb{Q}$, but I wonder does that holds for number field $\Bbb{Q}(\sqrt{-3})$.
I calculated its Tate-Shafarevich group and proved the equation has solution in $\Bbb{Q}(\sqrt{-3})$, but having difficulty finding exactly what the solution is.
If it has rational points, results only or computer calculation is really appreciated. Thank you in advance.
P.S. Sorry for my mistake, Tate-Shafarevich group was not trivial and I (may) proved this has no solution by using Brauer-Manin obstruction.So I can close this question but some elementary answer may come, so I leave this question.