# Tensor product of exterior power

I am trying to use the SageManifolds tensor modules package to work with the tensor power $T = E^{\otimes 2}$ where $E$ is itself the exterior power $E = \bigwedge^2 \mathbb{Z}$.

Here is what I have so far:

```
sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
sage: E = M.exterior_power(2)
sage: T = E.tensor_module(2,0)
```

I then want to work with an element of $T$ by giving it coordinates. I can give an element of $E$ a coordinate by doing

```
sage: a = E([], name='a')
sage: a.set_comp()[0,1] = 3
sage: a.set_comp()[0,2] = 1
sage: a.display()
a = 3 e_0∧e_1 + e_0∧e_2
```

So I would have expected to be able to do the same for $T$:

```
sage: t = T([], name='t')
sage: t.set_comp()[0,0] = 1
```

but I get a `ValueError`

:

```
ValueError: the None has not been defined on the 2nd exterior power of the Rank-3 free module M over the Integer Ring
```

The code in the source which prints `None`

is `raise ValueError("the {} has not been ".format(basis) +...`

and so the problem is that there is no basis for `E`

.

```
sage: E.default_basis()
No default basis has been defined on the 2nd exterior power of the Rank-3 free module M over the Integer Ring
```

When I try to define a basis I get a type error:

```
sage: E.basis('e')
TypeError: __init__() missing 1 required positional argument: 'degree'
```

coming from the line

```
sage/tensor/modules/free_module_basis.py in __init__(self, fmodule, symbol, latex_symbol, indices, latex_indices, symbol_dual, latex_symbol_dual)
637 ring_one = fmodule._ring.one()
638 for i in fmodule.irange():
--> 639 v = fmodule.element_class(fmodule)
640 v.set_comp(self)[i] = ring_one
641 vl.append(v)
```

because `E.element_class()`

has the init signature `E.element_class(fmodule, degree, name=None, latex_name=None)`

. There is no default degree passed, and there is no option for `fmodule.element_class(fmodule)`

to take an argument `degree`

. Are there any workarounds to this problem?

Many thanks!!