Elliptic Curves on a Small Lattice
A cubic equation is of the form . Given any nine lattice points, a cubic equation can be found whose plot, an elliptic curve, goes through all nine points, as shown in the "Nine-Point Cubic" Demonstration. More than nine lattice points can be covered, even when the lattice is tightly restricted.[more]
If a secant (or nontangent) line is drawn through two rational points on an elliptic curve, it also passes through a third rational point. Integer points are also rational, so it is possible to get a lot of "three-in-a-row" examples with an elliptic curve.
In this Demonstration, more than a thousand elliptic curves were chosen that visit many lattice points; they are roughly ranked by the number of lines they produce.[less]