Short Weierstrass curves

An elliptic curve in short Weierstrass form [database entry; Sage verification script; Sage output] has parameters a b and coordinates x y satisfying the following equations:

  y^2=x^3+a*x+b

The neutral element of the curve is the unique point at infinity, namely (0:1:0) in projective coordinates.

Representations for fast computations

Jacobian coordinates with a4=-3 [more information] make the additional assumptions

  a=-3

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

  x=X/Z^2
  y=Y/Z^3

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

  x=X/Z^2
  y=Y/Z^3

Projective coordinates with a4=-1 [more information] make the additional assumptions

  a=-1

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

  x=X/Z
  y=Y/Z

Projective coordinates with a4=-3 [more information] make the additional assumptions

  a=-3

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

  x=X/Z
  y=Y/Z

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

  x=X/Z
  y=Y/Z

XYZZ coordinates with a4=-3 [more information] make the additional assumptions

  a=-3

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

  x=X/ZZ
  y=Y/ZZZ
  ZZ^3=ZZZ^2

XYZZ coordinates [more information] represent x y as X Y ZZ ZZZ satisfying the following equations:

  x=X/ZZ
  y=Y/ZZZ
  ZZ^3=ZZZ^2