It turns out that for some elliptic curves over , such as , there are only finitely many rational points, while for others, such as the example above, there are infinitely many.
But in any case there is a deep theorem, proved by Mordell, that says that the group of rational points on an elliptic curve is ``finitely generated.'' This means there is a finite list of rational points on such that all rational points on can be generated from the points in by iteratively applying to pairs of points.
On the other hand, it is not known whether there exists an algorithm that takes the equation of an elliptic curve and outputs a finite list of generating points as above. Researchers in number theory have spent about 70 years trying to prove the existence of such an algorithm, but the problem is still unsolved!