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

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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

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(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:$x^{9} - 24 x^{7} - 23 x^{6} + 135 x^{5} + 159 x^{4} - 209 x^{3} - 180 x^{2} + 141 x + 11$