# Properties

 Label 3.12.12.22 Base $$\Q_{3}$$ Degree $$12$$ e $$3$$ f $$4$$ c $$12$$ Galois group $S_3\times C_3:S_3.C_2$ (as 12T119)

# Related objects

## Defining polynomial

 $$x^{12} + 18 x^{11} + 21 x^{10} - 69 x^{9} - 81 x^{8} + 72 x^{7} - 90 x^{6} - 108 x^{5} + 54 x^{4} - 108 x^{3} - 81$$

## Invariants

 Base field: $\Q_{3}$ Degree $d$: $12$ Ramification exponent $e$: $3$ Residue field degree $f$: $4$ Discriminant exponent $c$: $12$ Discriminant root field: $\Q_{3}(\sqrt{2})$ Root number: $-1$ $|\Aut(K/\Q_{ 3 })|$: $1$ 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: 3.4.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of $$x^{4} - x + 2$$ Relative Eisenstein polynomial: $$x^{3} + 3 t x^{2} + \left(-3 t^{2} - 3\right) x - 3 t^{3} + 3 t^{2} + 3$$$\ \in\Q_{3}(t)[x]$

## Invariants of the Galois closure

 Galois group: $S_3\times C_3:S_3.C_2$ (as 12T119) Inertia group: Intransitive group isomorphic to $C_3^3:C_2$ Wild inertia group: $C_3^3$ Unramified degree: $4$ Tame degree: $2$ Wild slopes: [3/2, 3/2, 3/2] Galois mean slope: $79/54$ Galois splitting model: $x^{12} + 12 x^{10} - 8 x^{9} + 54 x^{8} - 72 x^{7} + 88 x^{6} - 216 x^{5} - 39 x^{4} - 152 x^{3} - 180 x^{2} + 192 x + 124$