Explicit-Formulas Database

Genus-1 curves over large-characteristic fields
# Jacobi quartics

An elliptic curve in Jacobi quartic form
[database entry;
Sage verification script;
Sage output]
has parameters
a
and coordinates
x
y
satisfying the following equations:
y^^{2}=x^^{4}+2*a*x^^{2}+1

Affine addition formulas: (x1,y1)+(x2,y2)=(x3,y3) where
x3 = (x1*y2+y1*x2)/(1-(x1*x2)^^{2})
y3 = ((1+(x1*x2)^^{2})*(y1*y2+2*a*x1*x2)+2*x1*x2*(x1^^{2}+x2^^{2}))/(1-(x1*x2)^^{2})^^{2}

Affine doubling formulas: 2(x1,y1)=(x3,y3) where
x3 = (x1*y1+y1*x1)/(1-(x1*x1)^^{2})
y3 = ((1+(x1*x1)^^{2})*(y1*y1+2*a*x1*x1)+2*x1*x1*(x1^^{2}+x1^^{2}))/(1-(x1*x1)^^{2})^^{2}

Affine negation formulas: -(x1,y1)=(-x1,y1).
The neutral element of a Jacobi quartic is the point (0,1).

## Representations for fast computations

Doubling-oriented XXYZZ coordinates
[more information]
make the additional assumptions
a^^{2}+c^^{2}=1

and
represent
x
y
as
X
XX
Y
Z
ZZ
satisfying the following equations:
x=X/Z
y=Y/ZZ
XX=X^^{2}
ZZ=Z^^{2}

Doubling-oriented XXYZZR coordinates
[more information]
make the additional assumptions

a^^{2}+c^^{2}=1

and
represent
x
y
as
X
XX
Y
Z
ZZ
R
satisfying the following equations:
x=X/Z
y=Y/ZZ
XX=X^^{2}
ZZ=Z^^{2}
R=2*X*Z

Doubling-oriented XYZ coordinates
[more information]
make the additional assumptions

a^^{2}+c^^{2}=1

and
represent
x
y
as
X
Y
Z
satisfying the following equations:
x=X/Z
y=Y/Z^^{2}

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

x=X/Z
y=Y/ZZ
XX=X^^{2}
ZZ=Z^^{2}

XXYZZR coordinates
[more information]
represent
x
y
as
X
XX
Y
Z
ZZ
R
satisfying the following equations:

x=X/Z
y=Y/ZZ
XX=X^^{2}
ZZ=Z^^{2}
R=2*X*Z

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

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