Another example of rational points on an elliptic curve

Here's another example of finding rational points of finite order on an elliptic curve.

Suppose that we have the elliptic curve y² = x³ - 2x + 1. In this case we have that D = 5, so that the possibilities for the y-coordinate of a rational point of finite order are limited to 0, ±1 by the Nagell-Lutz theorem. A quick check of these possibilities shows that we have the following points:

P₁ = (1,0)

P₂ = (0,1)

P₃ = (0,-1)

Here's what this looks like:

These points form this subgroup of the points on the curve:

Rational points on y² = x³ - 2x + 1

+	O	P1	P2	P3
O	O	P1	P2	P3
P1	P1	O	P3	P2
P2	P2	P3	P1	O
P3	P3	P2	O	P1

We can also think of the elliptic curve y² = x³ - 2x + 1 as being parameterized by the Weierstrass ℘-function with periods ω₁ and ω₂ where we have approximately

ω₁ = -2.01891 i

and

ω₂ = 2.96882

All of the points on y² = x³ - 2x + 1 that have the property 4P = O come from the complex numbers shown here: 

 
Of these 16 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):