Which is isomorphic to $2y^2=x^4-17$ ?, $y^2=-34x^4+2$ or $y^2=-34x^4-2$

Question

I heard curve $2y^2=x^4-17$ is isomorphic over $\Bbb{Q}$ to one of the two curve, $y^2=-34x^4+2$ or $y^2=-34x^4-2$.

But I'm stucking which curve is isomorphic to $2y^2=x^4-17$. Those tree curves are isomorphic over $\Bbb{Q}(\sqrt{-2})$.

What is the strategy to determine which curve is isomorphic to $2y^2=x^4-17$ ?

Reference: Silverman's book 'The arithmetic of elliptic curves'', chapter 10

Answer 1

Under the change of variables $x\mapsto 1/x,y\mapsto y/2x^2$, the equation $2y^2=x^4-17$ becomes \begin{align*} 2\big(\frac y{2x^2}\big)^2&=x^{-4}-17\\ y^2&=2-34x^4. \end{align*} Conversely, it is not isomorphic to $y^2=-34x^4-2$, since it has no $\mathbb R$-points.