Rank

The rank of an elliptic curve is the largest number of rational points on it which are linearly independent with respect to its point group operation.

This is also the smallest number of rational points on the elliptic curve which generate all the others.

Mordell proved by a method of heights and descent that all elliptic curves with rational coefficients have finite rank.

There is said to be an isomorphism of abelian groups

$E(K)\cong \mathbb {Z} ^{r}\oplus E(K)_{\text{tors}}$

Computing the rank

The rank of randomly chosen elliptic curves is claimed to be very small, 0 or 1, on average about 1/2, although curves of arbitrarily large rank are conjectured to exist.

See Birch and Swinnerton-Dyer conjecture.