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:

  y^2=x^4+2*a*x^2+1

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

  a^2+c^2=1

and represent x y as X XX Y Z ZZ satisfying the following equations:

  x=X/Z
  y=Y/ZZ
  XX=X^2
  ZZ=Z^2

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

  a^2+c^2=1

and represent x y as X XX Y Z ZZ R satisfying the following equations:

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

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

  a^2+c^2=1

and represent x y as X Y Z satisfying the following equations:

  x=X/Z
  y=Y/Z^2

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

  x=X/Z
  y=Y/ZZ
  XX=X^2
  ZZ=Z^2

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=X^2
  ZZ=Z^2
  R=2*X*Z

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

  x=X/Z
  y=Y/Z^2