# Properties

 Label 3.12.18.82 Base $$\Q_{3}$$ Degree $$12$$ e $$6$$ f $$2$$ c $$18$$ Galois group $C_6\times C_2$ (as 12T2)

# Related objects

## Defining polynomial

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

## Invariants

 Base field: $\Q_{3}$ Degree $d$ : $12$ Ramification exponent $e$ : $6$ Residue field degree $f$ : $2$ Discriminant exponent $c$ : $18$ 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: $\Q_{3}(\sqrt{*})$ $\cong \Q_{3}(t)$ where $t$ is a root of $$x^{2} - x + 2$$ Relative Eisenstein polynomial: $x^{6} + \left(-6 t + 3\right) x^{5} + \left(-6 t + 12\right) x^{4} + \left(-6 t + 3\right) x^{3} - 9 x^{2} - 9 x + 12 t + 3 \in\Q_{3}(t)[x]$

## Invariants of the Galois closure

 Galois group: $C_2\times C_6$ (as 12T2) Inertia group: Intransitive group isomorphic to $C_6$ Unramified degree: $2$ Tame degree: $2$ Wild slopes: [2] Galois mean slope: $3/2$ Galois splitting model: $x^{12} - x^{6} + 1$