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
```
```  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

```  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
```

```  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
```

```  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
```