# Properties

 Label 3.12.14.15 Base $$\Q_{3}$$ Degree $$12$$ e $$6$$ f $$2$$ c $$14$$ Galois group $C_6\times S_3$ (as 12T18)

# Related objects

## Defining polynomial

 $$x^{12} + 9 x^{11} - 6 x^{10} + 6 x^{9} - 3 x^{8} + 9 x^{7} + 6 x^{6} - 9 x^{5} - 9 x^{4} - 9 x^{3} - 9 x^{2} + 9$$

## Invariants

 Base field: $\Q_{3}$ Degree $d$ : $12$ Ramification exponent $e$ : $6$ Residue field degree $f$ : $2$ Discriminant exponent $c$ : $14$ Discriminant root field: $\Q_{3}$ Root number: $1$ $|\Aut(K/\Q_{ 3 })|$: $6$ 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: $x^{6} + \left(-3 t - 3\right) x^{5} - 3 x^{4} - 3 t x^{3} + 3 x^{2} + 3 \in\Q_{3}(t)[x]$

## Invariants of the Galois closure

 Galois group: $C_6\times S_3$ (as 12T18) Inertia group: Intransitive group isomorphic to $S_3$ Unramified degree: $6$ Tame degree: $2$ Wild slopes: [3/2] Galois mean slope: $7/6$ Galois splitting model: $x^{12} - 2 x^{9} + 29 x^{6} - 28 x^{3} + 7$