rank of an elliptic curve
Let $K$ be a number field^{} and let $E$ be an elliptic curve^{} over $K$. By $E(K)$ we denote the set of points in $E$ with coordinates in $K$.
Theorem 1 (MordellWeil).
$E(K)$ is a finitely generated^{} abelian group^{}.
Proof.
The proof of this theorem is fairly involved. The
main two ingredients are the so called “weak MordellWeil theorem”
(see below), the concept of height function for abelian groups and
the “descent” theorem.
See [2], Chapter VIII, page
189.
∎
Theorem 2 (Weak MordellWeil).
$E(K)/mE(K)$ is finite for all $m\mathrm{\ge}\mathrm{2}$.
The MordellWeil theorem^{} implies that for any elliptic curve $E/K$ the group of points has the following structure:
$$E(K)\simeq {E}_{\mathrm{torsion}}(K)\oplus {\mathbb{Z}}^{R}$$ 
where ${E}_{\mathrm{torsion}}(K)$ denotes the set of points of finite order (or torsion group^{}), and $R$ is a nonnegative integer which is called the $rank$ of the elliptic curve. It is not known how big this number $R$ can get for elliptic curves over $\mathbb{Q}$. The largest rank known for an elliptic curve over $\mathbb{Q}$ is 28 http://www.math.hr/ duje/tors/tors.htmlElkies (2006).
Note: see Mazur’s theorem for an account of the possible torsion subgroups over $\mathbb{Q}$.
Examples:

1.
The elliptic curve ${E}_{1}/\mathbb{Q}:{y}^{2}={x}^{3}+6$ has rank 0 and ${E}_{1}(\mathbb{Q})\simeq 0$.

2.
Let ${E}_{2}/\mathbb{Q}:{y}^{2}={x}^{3}+1$, then ${E}_{2}(\mathbb{Q})\simeq \mathbb{Z}/6\mathbb{Z}$. The torsion group is generated by the point $(2,3)$.

3.
Let ${E}_{3}/\mathbb{Q}:{y}^{2}={x}^{3}+109858299531561$, then ${E}_{3}(\mathbb{Q})\simeq \mathbb{Z}/3\mathbb{Z}\oplus {\mathbb{Z}}^{5}$. See http://math.bu.edu/people/alozano/Torsion.htmlgenerators^{} here.

4.
Let ${E}_{4}/\mathbb{Q}:{y}^{2}+1951/164xy3222367/40344y={x}^{3}+3537/164{x}^{2}40302641/121032x$, then ${E}_{4}(\mathbb{Q})\simeq {\mathbb{Z}}^{10}$. See http://math.bu.edu/people/alozano/Examples.htmlgenerators here.
References
 1 James Milne, Elliptic Curves, online course notes. http://www.jmilne.org/math/CourseNotes/math679.htmlhttp://www.jmilne.org/math/CourseNotes/math679.html
 2 Joseph H. Silverman, The Arithmetic of Elliptic Curves. SpringerVerlag, New York, 1986.
 3 Joseph H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves. SpringerVerlag, New York, 1994.
 4 Goro Shimura, Introduction to the Arithmetic Theory of Automorphic Functions. Princeton University Press, Princeton, New Jersey, 1971.
Title  rank of an elliptic curve 
Canonical name  RankOfAnEllipticCurve 
Date of creation  20130322 13:49:12 
Last modified on  20130322 13:49:12 
Owner  alozano (2414) 
Last modified by  alozano (2414) 
Numerical id  14 
Author  alozano (2414) 
Entry type  Definition 
Classification  msc 14H52 
Synonym  rank 
Related topic  EllipticCurve 
Related topic  HeightFunction 
Related topic  MordellWeilTheorem 
Related topic  SelmerGroup 
Related topic  MazursTheoremOnTorsionOfEllipticCurves 
Related topic  NagellLutzTheorem 
Related topic  ArithmeticOfEllipticCurves 
Defines  weak MordellWeil theorem 
Defines  rank of an elliptic curve 