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

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

Affine addition formulas: (x1,y1)+(x2,y2)=(x3,y3) where

 x3 = (y2-y1)^2/(x2-x1)^2-x1-x2
 y3 = (2*x1+x2)*(y2-y1)/(x2-x1)-(y2-y1)^3/(x2-x1)^3-y1

Affine doubling formulas: 2(x1,y1)=(x3,y3) where

 x3 = (3*x1^2+a)^2/(2*y1)^2-x1-x1
 y3 = (2*x1+x1)*(3*x1^2+a)/(2*y1)-(3*x1^2+a)^3/(2*y1)^3-y1

Affine negation formulas: -(x1,y1)=(x1,-y1).

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=0 [more information] make the additional assumptions

 a=0

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

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

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

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

 x=X/Z^2
 y=Y/Z^3
 T=a*Z^4

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

W12 coordinates with a6=0 [more information] make the additional assumptions

 b=0

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

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

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

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

 x=X/Z