# Properties

 Label 3.12.12.28 Base $$\Q_{3}$$ Degree $$12$$ e $$6$$ f $$2$$ c $$12$$ Galois group $S_3\wr C_2$ (as 12T34)

# Learn more about

## Defining polynomial

 $$x^{12} + 12 x^{11} - 3 x^{10} + 3 x^{9} + 3 x^{8} + 6 x^{7} + 12 x^{6} + 9 x^{5} + 9 x^{4} + 9 x + 9$$

## Invariants

 Base field: $\Q_{3}$ Degree $d$ : $12$ Ramification exponent $e$ : $6$ Residue field degree $f$ : $2$ Discriminant exponent $c$ : $12$ Discriminant root field: $\Q_{3}$ Root number: $1$ $|\Aut(K/\Q_{ 3 })|$: $2$ This field is not Galois over $\Q_{3}$.

## Intermediate fields

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

## Unramified/totally ramified tower

 Unramified subfield: $\Q_{3}(\sqrt{*})$ $\cong \Q_{3}(t)$ where $t$ is a root of $$x^{2} - x + 2$$ Relative Eisenstein polynomial: $x^{6} - 3 t x^{5} + \left(-3 t - 3\right) x^{4} - 3 x^{3} + \left(3 t + 3\right) x^{2} - 3 x - 3 \in\Q_{3}(t)[x]$

## Invariants of the Galois closure

 Galois group: $S_3\wr C_2$ (as 12T34) Inertia group: Intransitive group isomorphic to $C_3:S_3.C_2$ Unramified degree: $2$ Tame degree: $4$ Wild slopes: [5/4, 5/4] Galois mean slope: $43/36$ Galois splitting model: $x^{12} - 6 x^{8} - 16 x^{6} + 57 x^{4} - 48 x^{2} + 16$