Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 21 / 2 / c / Solution

The rational point $P=(0,1)$ lies on $E$ and is neither the identity nor $(-1,0)$. By solution these are all the rational torsion points of an elliptic curve. Hence $P$ has infinite order and $E(\mathbb{Q})$ is infinite: its multiples $nP$ are distinct. No assertion about its being a generator or about the exact rank of an elliptic curve is needed here.