The abelian -commutative- group defined on elliptic curves [Le groupe abélien -commutatif- défini sur les courbes elliptiques].

                    y2 = x3 - x + 1

• 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]

                            1       1
                    P = {- --- , + ---}
                            1       1

                            1       7
                    Q = {+ --- , + ---}
                            4       8

                            19       103
                    R = {+ ---- , + -----}
                            25       125