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:

Image001

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: 

Lattice4 
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):

Lattice3

Leave a Reply

Your email address will not be published. Required fields are marked *