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:

  s^2+c^2=1
  a*s^2+d^2=1

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