Properties

Label 3.9.12.17
Base \(\Q_{3}\)
Degree \(9\)
e \(3\)
f \(3\)
c \(12\)
Galois group $C_3^2 : S_3 $ (as 9T13)

Related objects

Learn more about

Defining polynomial

\( x^{9} + 6 x^{8} + 6 x^{6} + 27 \)

Invariants

Base field: $\Q_{3}$
Degree $d$ : $9$
Ramification exponent $e$ : $3$
Residue field degree $f$ : $3$
Discriminant exponent $c$ : $12$
Discriminant root field: $\Q_{3}(\sqrt{*})$
Root number: $1$
$|\Aut(K/\Q_{ 3 })|$: $1$
This field is not Galois over $\Q_{3}$.

Intermediate fields

3.3.0.1

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(3 t^{2} + 6 t\right) x^{2} + 3 t^{2} + 6 t \in\Q_{3}(t)[x]$

Invariants of the Galois closure

Galois group:$He_3:C_2$ (as 9T13)
Inertia group:Intransitive group isomorphic to $C_3^2$
Unramified degree:$6$
Tame degree:$1$
Wild slopes:[2, 2]
Galois mean slope:$16/9$
Galois splitting model:$x^{9} + 3 x^{7} - 6 x^{6} - 18 x^{5} - 12 x^{4} - 22 x^{3} - 27 x^{2} + 12 x + 13$