# Properties

 Label 3.12.12.17 Base $$\Q_{3}$$ Degree $$12$$ e $$3$$ f $$4$$ c $$12$$ Galois group 12T170

# Learn more about

## Defining polynomial

 $$x^{12} + 12 x^{11} - 120 x^{10} + 108 x^{9} + 90 x^{8} + 99 x^{7} + 90 x^{6} - 27 x^{5} + 54 x^{4} + 81 x^{3} - 81 x^{2} - 81$$

## Invariants

 Base field: $\Q_{3}$ Degree $d$ : $12$ Ramification exponent $e$ : $3$ Residue field degree $f$ : $4$ Discriminant exponent $c$ : $12$ Discriminant root field: $\Q_{3}(\sqrt{*})$ 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: $x^{3} + \left(-3 t^{3} - 3\right) x^{2} + \left(3 t^{3} - 3 t^{2} - 3\right) x + 3 t^{3} - 3 t^{2} - 3 t + 3 \in\Q_{3}(t)[x]$

## Invariants of the Galois closure

 Galois group: 12T170 Inertia group: Intransitive group isomorphic to $C_3:(C_3^3:C_2)$ Unramified degree: $4$ Tame degree: $2$ Wild slopes: [3/2, 3/2, 3/2, 3/2] Galois mean slope: $241/162$ Galois splitting model: $x^{12} - 8 x^{9} + 16 x^{6} - 8 x^{3} - 2$