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32 1. Elementary Properties of the Chord-Tangent Group Law on a Cubic Curve

(u,v,w) = (0, 0, 0), (1, 0, 1), (−1, 0, −1), (0, 1, 1),and (0, −1, −1). This is not
even a cubic equation, but under the transformation

2yw 2 1
2
u = and v = w 1 − ,
x 2 x
we obtain
4 2
4w y 1 4
4 4
w = + w 1 − .
x 4 x
4
4
Dividing by w and multiplying by x , we derive the cubic equation
3 2 1
2
3
y = x − x + x − .
2 4
The equation factors giving one point (1/2, 0) of order 2

1 1
2 2
y = x − x − x + .
2 2
Consider the lines y = λ(x−1/2) through (1/2, 0) and their other intersection points
with the cubic. They have x-coordinates satisfying
1 2 1

2
λ 1 − = x − x +
2 2
or equivalently
1 2
2
2
0 = x − (1 + λ )x + (1 + λ ).
2
The two other intersection points will coincide, that is, the line will be tangent to
curve if and only if
2 2
2
(1 + λ ) = 2(1 + λ ).
2
2
Since λ + 1 is nonzero, we can divide by it, and we obtain λ = 1or λ =+1, −1.
Substituting back for
2
2
1. λ = 1, y = x − 1/2sothat0 = x − 2x + 1 = (x − 1) and (1, 1/2) is a point
on the curve with 2(1, 1/2) = (1/2, 0),and
2. λ =−1, gives by the same argument (1, −1/2) with 2(1, −1/2) = (1/2, 0),
but this could have been deduced immediately from (1) by 2(1, −1/2) =
−2(1, 1/2) =−(1/2, 0) = (1/2, 0). It is an application of the group structure.
For this curve E with equation in normal form
3 2 1
3
2
y = x − x + x − ,
2 4