Explicit-Formulas Database
Genus-1 curves over large-characteristic fields

Montgomery curves

An elliptic curve in Montgomery form [database entry; Sage verification script; Sage output] has parameters a b and coordinates x y satisfying the following equations:
Affine addition formulas: (x1,y1)+(x2,y2)=(x3,y3) where
  x3 = b*(y2-y1)2/(x2-x1)2-a-x1-x2
  y3 = (2*x1+x2+a)*(y2-y1)/(x2-x1)-b*(y2-y1)3/(x2-x1)3-y1
Affine doubling formulas: 2(x1,y1)=(x3,y3) where
  x3 = b*(3*x12+2*a*x1+1)2/(2*b*y1)2-a-x1-x1
  y3 = (2*x1+x1+a)*(3*x12+2*a*x1+1)/(2*b*y1)-b*(3*x12+2*a*x1+1)3/(2*b*y1)3-y1
Affine negation formulas: -(x1,y1)=(x1,-y1).

Representations for fast computations

XZ coordinates [more information] represent x y as X Z satisfying the following equations: