More interesting elliptic curve stuff

It turns out that we can easily tell some things about an elliptic curve of the form

y2 = 4 x3g2 x – g3

Suppose that we get this elliptic curve from the Weierstrass ℘-function with periods ω1 and ω2. Then it turns out that we we can also write this elliptic curve as

y2 = 4 (xx1)(xx2)(xx3)

where

x1 = ℘(ω1 / 2)

x2 = ℘(ω2 / 2)

and

x3 = ℘((ω1 + ω2) / 2)

For an example of this, consider the elliptic curve

y2 = 4 (x – 1)(x – 2)(x + 3)

= 4 x3 – 28 x + 24

which we get from the Weierstrass ℘-function with periods approximately

ω1 = -1.48441 i

and

ω2 = 2.01891

In this case, we find that 

℘(ω1 / 2) = -3

℘(ω2 / 2) = 2

and

℘((ω1 + ω2) / 2) = 1

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