# Properties

 Label 3.12.6.2 Base $$\Q_{3}$$ Degree $$12$$ e $$2$$ f $$6$$ c $$6$$ Galois group $C_6\times C_2$ (as 12T2)

# Related objects

## Defining polynomial

 $$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$

## Invariants

 Base field: $\Q_{3}$ Degree $d$ : $12$ Ramification exponent $e$ : $2$ Residue field degree $f$ : $6$ Discriminant exponent $c$ : $6$ Discriminant root field: $\Q_{3}$ Root number: $1$ $|\Gal(K/\Q_{ 3 })|$: $12$ This field is Galois and abelian 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: 3.6.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of $$x^{6} - x + 2$$ Relative Eisenstein polynomial: $x^{2} + \left(-3 t^{3} - 3 t + 3\right) x - 3 t^{4} - 3 t^{3} + 3 t^{2} + 3 \in\Q_{3}(t)[x]$

## Invariants of the Galois closure

 Galois group: $C_2\times C_6$ (as 12T2) Inertia group: Intransitive group isomorphic to $C_2$ Unramified degree: $6$ Tame degree: $2$ Wild slopes: None Galois mean slope: $1/2$ Galois splitting model: $x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1$