# SAGEMATH module: superelliptic curves and their Artin-Schreier covers

## Basic information

## Usage

The main file is init.sage. In order to use it, type:

```sage: load('init.sage')```

The main two "packages" are intended for:
- superelliptic curves,
- $(\mathbb Z/p)^n$-covers of superelliptic curves.

## Superelliptic curves

In order to define a superelliptic curve $C : y^4 = x^6 + 1$ over the finite field with 9 elements,
use the following commands:

```
F.<a> = GF(9, 'a')
Rx.<x> = PolynomialRing(F)
f = x^6 + 1
C = superelliptic(f, 4)
```
There are three auxilliary classes: superelliptic_function (for functions defined on superelliptic curves), superelliptic_form (for forms defined on superelliptic curves) and superelliptic_cech (for cech cocycles for the de Rham cohomology on superelliptic curves).

For example, in order to define the function $x + y$ on our curve $C$ we can define it like this:

```
Rxy.<x, y> = PolynomialRing(F, 2)
fct = superelliptic_function(C, x + y)
```
or simpler:
```
fct = C.x + C.y
```

Similarly, in order to define the form $\omega = y \cdot dx$ we may use:
```
omega = superelliptic_form(C, y)
```
or simpler:
```
omega = C.y * C.dx
```



## Troubleshooting

- precision
- threshold
- no root in the field
- basis -- coordinates.