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

Jacobi quartics

An elliptic curve in Jacobi quartic form [database entry; Sage verification script; Sage output] has parameters a and coordinates x y satisfying the following equations: 
  y2=x4+2*a*x2+1
Affine addition formulas: (x1,y1)+(x2,y2)=(x3,y3) where 
  x3 = (x1*y2+y1*x2)/(1-(x1*x2)2)
  y3 = ((1+(x1*x2)2)*(y1*y2+2*a*x1*x2)+2*x1*x2*(x12+x22))/(1-(x1*x2)2)2
Affine doubling formulas: 2(x1,y1)=(x3,y3) where 
  x3 = (x1*y1+y1*x1)/(1-(x1*x1)2)
  y3 = ((1+(x1*x1)2)*(y1*y1+2*a*x1*x1)+2*x1*x1*(x12+x12))/(1-(x1*x1)2)2
Affine negation formulas: -(x1,y1)=(-x1,y1).

The neutral element of a Jacobi quartic is the point (0,1).

Representations for fast computations

Doubling-oriented XXYZZ coordinates [more information] make the additional assumptions 
  a2+c2=1
and represent x y as X XX Y Z ZZ satisfying the following equations: 
  x=X/Z
  y=Y/ZZ
  XX=X2
  ZZ=Z2

Doubling-oriented XXYZZR coordinates [more information] make the additional assumptions 

  a2+c2=1
and represent x y as X XX Y Z ZZ R satisfying the following equations: 
  x=X/Z
  y=Y/ZZ
  XX=X2
  ZZ=Z2
  R=2*X*Z

Doubling-oriented XYZ coordinates [more information] make the additional assumptions 

  a2+c2=1
and represent x y as X Y Z satisfying the following equations: 
  x=X/Z
  y=Y/Z2

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

  x=X/Z
  y=Y/ZZ
  XX=X2
  ZZ=Z2

XXYZZR coordinates [more information] represent x y as X XX Y Z ZZ R satisfying the following equations: 

  x=X/Z
  y=Y/ZZ
  XX=X2
  ZZ=Z2
  R=2*X*Z

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

  x=X/Z
  y=Y/Z2