Parameterizing elliptic curves

Here's an example of parameterizing an elliptic curve of the form

y2 = 4 x3g2 x – g3

using the Weierstrass ℘-function.

Consider the elliptic curve

y2 = 4 (x – 1)(x – 2)(x + 3)

= 4 x3 – 28 x + 24

This is a handy elliptic curve to use because we can easily see where its graph crosses the x-axis, etc. Identifying this with

℘′(z)2 = 4 ℘(z)3g2 ℘(z) – g3

we have that

g2 = 28

and

g3 = -24

To plot the graph of this elliptic curve when it's parameterized as (℘(z), ℘′(z)) we first need to find the periods ω1 and ω2, which we can find from g2 and g3. Those turn out to be approximately

ω1 = 1.48441 i

and

ω2 = 2.01891

Here's what the magnitude of the corresponding ℘-function looks like over a few of its periods (graphed using Mathematica, as usual).

Image001

If we then plot the graph of (℘(z), ℘′(z)) as z goes along the real line from 0.7 to 1.4,  we get the following graph, which is the graph of y2 = 4 x3 – 28 x + 24:

Image001 

Another example

Doing the same thing for

g2 = 4

and

g3 = -4

and plotting the graph of (℘(z), ℘′(z)) as z goes along the real line from 0.7 to 4.0 we get this, which is the graph of y2 = 4 x3 - 4 x + 4:

Image001
That's definitely another elliptic curve.

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