Elliptic Curves on a Small Lattice

​
select elliptic curve
1
A cubic equation is of the form
a
0
+
a
1
x+
a
2
2
x
+
a
3
3
x
+
a
4
y+
a
5
xy+
a
6
2
x
y+
a
7
2
y
+
a
8
x
2
y
+
a
9
3
y
=0
. 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.
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.

External Links

Elliptic Curve (Wolfram MathWorld)
Nine-Point Cubic
Rational Points on an Elliptic Curve

Permanent Citation

Ed Pegg Jr
​
​"Elliptic Curves on a Small Lattice"​
​http://demonstrations.wolfram.com/EllipticCurvesOnASmallLattice/​
​Wolfram Demonstrations Project​
​Published: June 1, 2011