# Ideals of Index

## `IdealsOfIndex(O, N): RngOrd, RngIntElt -> SeqEnum[RngOrdIdl]`

Given an order $O$ in a number field and a positive integer $N$, returns all the ideals $I$ of index $[O:I]=N$.

## `IdealsOfIndex(I, N): RngOrdIdl, RngIntElt -> SeqEnum[RngOrdIdl]`

Given an ideal $I$ in an order $O$ in a number field and a positive integer $N$, with $N$ coprime with the conductor, returns all the ideals $J$ contained in $I$ with index $[I:J]=N$.

## `IdealsOfIndex(I, N): RngOrdFracIdl, RngIntElt -> SeqEnum[RngOrdFracIdl]`

Given an ideal $I$ in an order $O$ in a number field and a positive integer $N$, with $N$ coprime with the conductor, returns all the ideals $J$ contained in $I$ with index $[I:J]=N$.

## `IdealsOfIndex(I, N): AlgEtQIdl, RngIntElt -> SeqEnum[AlgEtQIdl]`

```magma
Method: MonStgElt                    Default: "Default"
```

Given an $O$-ideal $I$ in $O$ and a positive integer $N$, returns all the subideals $J$ of $I$ with index $[I:J]=N$. The function is very fast if $N$ is coprime to the conductor of $O$. If this condition is not satisfied a slow algorithm is used which doesn’t require additional hypothesis. One can force the slow algorithm by setting the parameter `Method:="Slow"`.

## `IdealsOfIndex(O, N): AlgEtQOrd, RngIntElt -> SeqEnum[AlgEtQIdl]`

```magma
Method: MonStgElt                    Default: "Default"
```

Given an order $O$ and a positive integer $N$, returns all the $O$-ideals $J$ with index $[O:J]=N$. The function is very fast if $N$ is coprime to the conductor of $O$. If this condition is not satisfied a slow algorithm is used which doesn’t require additional hypothesis. One can force the slow-naive algorithm by setting the parameter `Method:="Slow"`.