Rational points on elliptic curves

If you look at the rules for adding points on an elliptic curve, you'll see that the coordinates of P₃ = P₁ + P₂ are rational functions of the coordinates of P₁and P₂. This means that if P₁and P₂ have rational coordinates then P₃ also does because its coordinates are rational functions of these rational coordinates.

It turns out that rational points on an ellipic curve also have other interesting properties. In particular, rational points of finite order have to have integer coordinates and the y-coordinate of these points has to either be 0 or divide the discriminant of the cubic that defines the elliptic curve. This is the Nagell-Lutz theorem, which is the following:

Let y² = x³ + ax + b be an elliptic curve over the rationals with integer coefficients and let D = 4 a³ + 27 b². Then if P = (x_P,y_P) is a rational point of finite order then P has integer coordinates and either y_P = 0 or y_P²|D.

Example

Suppose that we have the elliptic curve y² = x³ + 1. In this case we have that D = -27, so that the possibilities for the y-coordinate of a rational point of finite order are limited to 0, ±1, ±3. A quick check of these possibilities shows that we have the following points, and the Nagell-Lutz theorem tells us that that's all of them:

P₁ = (-1,0)

P₂ = (0,1)

P₃ = (0,-1)

P₄ = (2,3)

P₅ = (2,-3)

Here's what this looks like:

 

It's also easy to check that these points plus O form a subgroup of the points on the curve. If you add any two of these points together you always get another of of these points. Here's this subgroup:

Rational points on y² = x³ + 1

+	O	P1	P2	P3	P4	P5
O	O	P1	P2	P3	P4	P5
P1	P1	O	P5	P4	P3	P2
P2	P2	P5	P3	O	P1	P4
P3	P3	P4	O	P2	P5	P1
P4	P4	P3	P1	P5	P2	O
P5	P5	P2	P4	P1	O	P3

We can also think of the elliptic curve y² = x³ + 1 as being parameterized by the Weierstrass ℘-function with periods ω₁ and ω₂ where we have approximately

ω₁ = 2.10327 + 1.21433 i

and

ω₂ = 2.42865 i

All of the points on y² = x³ + 1 that have the property 6P = O come from the complex numbers shown here: 

   

Of these 36 points, these are the ones that we get the subgroup of rational points of finite order from (z₁ corresponds to P₁, etc., and z₀ corresponds to O):