![](image.jpg)
The abelian -commutative- group defined on elliptic curves [Le groupe abélien -commutatif- défini sur les courbes elliptiques].
The continuous white line displays the following elliptic curve:
y2 = x3 - x + 1
An elliptic curve is an abelian -i.e. commutative- group:
- Definitions:
- I (the identity of the group) is the point at infinity (a vertical line on the picture).
- The inverse of a point P=P(x,y) is -P=P(x,-y). On the picture R' means -R and defines P+Q.
- P+Q+R = I
- Abelian group laws:
- P+I = I+P = P [Identity]
- P+(-P) = (-P)+P = I [Inverse]
- (P+Q)+R = P+(Q+R) [Associativity]
- P+Q = Q+P [Commutativity]
On this picture, the 3 points P, Q and R have rational coordinates:
1 1
P = {- --- , + ---}
1 1
1 7
Q = {+ --- , + ---}
4 8
19 103
R = {+ ---- , + -----}
25 125
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