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

Jacobi intersections

An elliptic curve in Jacobi intersection form [database entry; Sage verification script; Sage output] has parameters a and coordinates s c d satisfying the following equations:
  s2+c2=1
  a*s2+d2=1
Affine addition formulas: (s1,c1,d1)+(s2,c2,d2)=(s3,c3,d3) where
  s3 = (c2*s1*d2+d1*s2*c1)/(c22+(d1*s2)2)
  c3 = (c2*c1-d1*s2*s1*d2)/(c22+(d1*s2)2)
  d3 = (d1*d2-a*s1*c1*s2*c2)/(c22+(d1*s2)2)
Affine doubling formulas: 2(s1,c1,d1)=(s3,c3,d3) where
  s3 = (c1*s1*d1+d1*s1*c1)/(c12+(d1*s1)2)
  c3 = (c1*c1-d1*s1*s1*d1)/(c12+(d1*s1)2)
  d3 = (d1*d1-a*s1*c1*s1*c1)/(c12+(d1*s1)2)
Affine negation formulas: -(s1,c1,d1)=(-s1,c1,d1).

The neutral element of a Jacobi intersection is the point (0,1,1). The parameter a is required to be different from 0 and 1.

Representations for fast computations

Extended coordinates [more information] represent s c d as S C D Z SC DZ satisfying the following equations:
  s=S/Z
  c=C/Z
  d=D/Z
  SC=S*C
  DZ=D*Z

Projective coordinates [more information] represent s c d as S C D Z satisfying the following equations:

  s=S/Z
  c=C/Z
  d=D/Z