Perhaps one of the most interesting properties of an elliptic curve is that the points on an elliptic curve form a group. The key step in seeing this is to note that given any two points on an elliptic curve we can, in a natural way, define what it means to ``add" those two points together. On an intuitive level, when you add two points together you draw a line connecting them, and see where that line intersects the elliptic curve. If there is no third point of intersection then we say that those two points add to ``the point at infinity," which is the identity element in the group. Otherwise, if they intersect at a third point, , then we define that the ``addition" of those two points to be . (The reason that you need to switch the -coordinate is so that all of the group axioms come out correctly.)
Once you determine that the points do indeed form a group, then the natural question to ask is, what is this group? If we are considering our curve over then the story becomes less interesting. In this case the group is infinite, as given any one can find such that lies on the curve. It being infinite is not what makes it uninteresting, but it is the fact that the group is infinitely generated that makes it so. Because it is infinitely generated there is not much we can say about the group. In fact for any elliptic curve, the group is always either or . Where is the group formed by the points in the complex plane on the circle of radius 1 under multiplication.
Since the problem of has been solved, we turn our attention to what happens when we consider our curve over . In this case Mordell's theorem tells us that the group is finitely generated and hence that
The integer is called the rank of the elliptic curve.
It is a folklore conjecture that can be arbitrarily large, however, the current record is a curve of rank at least 24. This was discovered by Roland Martin and William McMillen of the National Security Agency in January 2000.
(See http://listserv.nodak.edu/scripts/wa.exe?A2=ind0005&L=nmbrthry&P=R182)