I am working through an exercise (below) in the book 'Diophantine Geometry: An Introduction' which asserts that the curve $2 Y^2 = X^4 - 17$ has two points at infinity, neither of which are $\mathbb{Q}$-rational.
This is incorrect, right... or am I missing something here?!
Homogenizing, we get $2Y^2 Z^2 = X^4 - 17Z^4$, so setting $Z=0$ gives $X = 0$, and hence, the unique point at infinity should be $[0; 1; 0]$ (which is rational over $\mathbb{Q}$)...
I would appreciate if someone could just verify this for me.