# Associated Spaces

## `BrandtModule(M): ModSS -> ModBrdt`

The Brandt module associated to the supersingular module $M$.

## `ModularSymbols(M : parameters): ModSS -> ModSym`

```magma
Proof: BoolElt                    Default: true
```

The space of modular symbols corresponding to the supersingular module $M$.

## `ModularSymbols(M, sign : parameters): ModSS, RngIntElt -> ModSym`

```magma
Proof: BoolElt                    Default: true
```

The $+1$ or $-1$ quotient of the space of modular symbols corresponding to the supersingular module $M$.

## `RSpace(M): ModSS -> ModTupRng, Map`

The ${\mathbb{Z}}$-module $V$ underlying the supersingular module $M$ along with an invertible map $V \rightarrow M$.

## `Example: Associated (ex-16e86e)`

We compute the Brandt module and modular symbols spaces associated to the supersingular module for $p=3$, $N=11$, and verify that $T_2$ has the same characteristic polynomial on each.

```magma
> M := SupersingularModule(3,11);
> B := BrandtModule(M); B;
Brandt module of level (3,11), dimension 2, and degree 2 over
Integer Ring
> MS := ModularSymbols(M); MS;
Modular symbols space for Gamma_0(33) of weight 2 and dimension 4
over Rational Field
> Factorization(CharacteristicPolynomial(HeckeOperator(B, 2)));
[
    <$.1 - 3, 1>,
    <$.1 - 1, 1>
]
> Factorization(CharacteristicPolynomial(HeckeOperator(MS, 2)));
[
    <$.1 - 3, 2>,
    <$.1 - 1, 2>
]

```

There is an associated Brandt module even if the underlying computations on $M$ are done using the Mestre-Oesterle graph method.

```magma
> M := SupersingularModule(11);
> UsesMestre(M);
true
> B := BrandtModule(M); B;   // takes a while
Brandt module of level (11,1), dimension 2, and degree 2 over
Integer Ring

```