# Properties

 Label 3.12.14.13 Base $$\Q_{3}$$ Degree $$12$$ e $$6$$ f $$2$$ c $$14$$ Galois group $C_3\times C_3:S_3.C_2$ (as 12T73)

# Related objects

## Defining polynomial

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

## 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: $-1$ $|\Aut(K/\Q_{ 3 })|$: $3$ 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: $x^{6} + \left(3 t + 3\right) x^{5} + \left(3 t + 3\right) x^{4} + 3 x^{3} + \left(-3 t + 3\right) x^{2} + 3 t \in\Q_{3}(t)[x]$

## Invariants of the Galois closure

 Galois group: $C_3\times C_3:S_3.C_2$ (as 12T73) Inertia group: Intransitive group isomorphic to $C_3:S_3$ Unramified degree: $6$ Tame degree: $2$ Wild slopes: [3/2, 3/2] Galois mean slope: $25/18$ Galois splitting model: $x^{12} + 84 x^{10} - 196 x^{9} + 2646 x^{8} - 12348 x^{7} + 54446 x^{6} - 259308 x^{5} + 925365 x^{4} - 2600528 x^{3} + 7674282 x^{2} - 16492812 x + 19131511$