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
```

(CMAP28 WWW site: this page was created on 09/18/2013 and last updated on 05/06/2022 14:37:02 -CEST-)

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