# Properties

 Label 3.12.14.14 Base $$\Q_{3}$$ Degree $$12$$ e $$6$$ f $$2$$ c $$14$$ Galois group $(C_3\times C_3):C_4$ (as 12T17)

# Related objects

## Defining polynomial

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

## Invariants

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

## Invariants of the Galois closure

 Galois group: $C_3:S_3.C_2$ (as 12T17) Inertia group: Intransitive group isomorphic to $C_3:S_3$ Unramified degree: $2$ Tame degree: $2$ Wild slopes: [3/2, 3/2] Galois mean slope: $25/18$ Galois splitting model: $x^{12} - 24 x^{8} - 12 x^{6} + 189 x^{4} + 324 x^{2} + 18$