# Elliptic Curves on a Small Lattice

Elliptic Curves on a Small Lattice

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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.

a+ax+ax+ax+ay+axy+axy+ay+axy+ay=0

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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.