Short Weierstrass curves

An elliptic curve in short Weierstrass form [database entry; Sage verification script; Sage output] has parameters a2 a6 and coordinates x y satisfying the following equations:

  y^2+x*y=x^3+a2*x^2+a6

Representations for fast computations

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

  x=X
  y=Y

Extended Lopez-Dahab coordinates with a2=0 [more information] make the additional assumptions

  a2=0

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

  x=X/Z
  y=Y/ZZ
  ZZ=Z^2
  XZ=X*Z

Extended Lopez-Dahab coordinates with a2=1 [more information] make the additional assumptions

  a2=1

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

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

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

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

Lopez-Dahab coordinates with a2=0 [more information] make the additional assumptions

  a2=0

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

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

Lopez-Dahab coordinates with a2=1 [more information] make the additional assumptions

  a2=1

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

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

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

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

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

  x=X/Z
  y=Y/Z