# Properties

 Label 3.9.12.4 Base $$\Q_{3}$$ Degree $$9$$ e $$3$$ f $$3$$ c $$12$$ Galois group $C_3^2:C_3$ (as 9T7)

# Related objects

## Defining polynomial

 $$x^{9} + 6 x^{6} + 18 x^{5} + 36 x^{3} + 27$$

## Invariants

 Base field: $\Q_{3}$ Degree $d$ : $9$ Ramification exponent $e$ : $3$ Residue field degree $f$ : $3$ Discriminant exponent $c$ : $12$ Discriminant root field: $\Q_{3}$ Root number: $1$ $|\Aut(K/\Q_{ 3 })|$: $3$ 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: 3.3.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of $$x^{3} - x + 1$$ Relative Eisenstein polynomial: $x^{3} + \left(18 t^{2} + 3 t + 6\right) x^{2} + 18 t^{2} x + 3 t^{2} + 18 t + 9 \in\Q_{3}(t)[x]$

## Invariants of the Galois closure

 Galois group: $He_3$ (as 9T7) Inertia group: Intransitive group isomorphic to $C_3^2$ Unramified degree: $3$ Tame degree: $1$ Wild slopes: [2, 2] Galois mean slope: $16/9$ Galois splitting model: $x^{9} - 24 x^{7} - 23 x^{6} + 135 x^{5} + 159 x^{4} - 209 x^{3} - 180 x^{2} + 141 x + 11$