# Properties

 Label 3.12.12.24 Base $$\Q_{3}$$ Degree $$12$$ e $$6$$ f $$2$$ c $$12$$ Galois group 12T215

# Related objects

## Defining polynomial

 $$x^{12} + 3 x^{7} + 3 x^{6} - 9 x^{2} + 9 x - 9$$

## Invariants

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

## Invariants of the Galois closure

 Galois group: 12T215 Inertia group: Intransitive group isomorphic to $C_3^2:(C_3:S_3.C_2)$ Unramified degree: $4$ Tame degree: $4$ Wild slopes: [5/4, 5/4, 5/4, 5/4] Galois mean slope: $403/324$ Galois splitting model: $x^{12} - 6 x^{11} + 12 x^{10} - 3 x^{9} - 15 x^{8} + 18 x^{7} - 13 x^{6} + 6 x^{5} + 21 x^{4} - 45 x^{3} + 15 x^{2} + 15 x - 5$