Given permutation module with three generators and corresponding Galois action of symmetric group $\mathfrak S_3$ I am interested in computing corresponding dual torus $T$ (which should be of dimension 3) over the ground field $k$.

I start with the group ring algebra $\overline k[x_1, x_2, x_3, \frac{1}{x_1}, \frac{1}{x_2}, \frac{1}{x_3}$] and look for the $\mathfrak S_3$-invariants there. In particular, the obvious ones are

$x_1 + x_2 + x_3$

$\frac{1}{x_1} + \frac{1}{x_2} + \frac{1}{x_3}$

$x_1x_2 + x_1x_3 + x_2x_3$

...

From the other side, there are certainly more since $\mathfrak S_3$ also acts on coefficients. In particular, if the splitting degree six extension is given by adjoining roots $\alpha_1, \alpha_2, \alpha_3$ of irreducible polynomial $y^3 - y + a$, where $a \in k$, then the following elements are invariants as well:

$\alpha_1x_1 + \alpha_2x_2 + (1 - \alpha_1 - \alpha_2) x_3$

$\frac{\alpha_1}{x_1} + \frac{\alpha_2}{x_2} + \frac{1-\alpha_1-\alpha_2}{x_3}$

...

My next step would be to write down relations between these generators, therefore, obtaining equations for the torus $T$. I coudn't proceed with this step as cannot see any good relations here. This approach worked well for the case of group $\mathfrak S_2$ though. Would be happy for any advice or reference as this problem looks like a classical one.