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 A=A(x,y) is -A=A(x,-y). On the picture C' means -C and defines A+B.
  • A+B+C = I


• Abelian group laws:
  
  
  • A+I = I+A = A [Identity]
  • A+(-A) = (-A)+A = I [Inverse]
  • (A+B)+C = A+(B+C) [Associativity]
  
  
  • A+B = B+A [Commutativity]

                            1       1
                    A = {- --- , + ---}
                            1       1

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

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