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

Twisted Hessian curves

An elliptic curve in twisted Hessian form [database entry; Sage verification script; Sage output] has parameters a d and coordinates x y satisfying the following equations:
Affine addition formulas: (x1,y1)+(x2,y2)=(x3,y3) where
  x3 = (x1-y12*x2*y2)/(a*x1*y1*x22-y2)
  y3 = (y1*y22-a*x12*x2)/(a*x1*y1*x22-y2)
Affine doubling formulas: 2(x1,y1)=(x3,y3) where
  x3 = (x1-y13*x1)/(a*y1*x13-y1)
  y3 = (y13-a*x13)/(a*y1*x13-y1)
Affine negation formulas: -(x1,y1)=(x1/y1,1/y1).

Representations for fast computations

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