Let $E$ be an elliptic curve defined over $\mathbb{Q}$. Is there an example of such an $E$ such that the only rational point in $E(\mathbb{Q})$ is the point at infinity?
In other words, consider the following Weierstrass equation: $$y^2 = x^3 + ax + b$$
Are there values of $a, b \in \mathbb{Q}$ (preferably in $\mathbb{Z}$) such that there is no $(x, y) \in \mathbb{Q}^2$ satisfying the equation?
As requested in the comments, I will upgrade my comment to an answer. Keith Conrad's excellent article settles the question. On page $10$, it is noted that for the values $k = -5, -6, 6, 7, -24, 45$, the Mordell curve $y^{2} = x^{3}+k$ has exactly one rational point. Proof of this fact is not given, so the references may be helpful in this regard. He does prove in each case that there are no integral points, however.