Properties

Label 3.12.12.2
Base \(\Q_{3}\)
Degree \(12\)
e \(3\)
f \(4\)
c \(12\)
Galois group 12T173

Related objects

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Defining polynomial

\( x^{12} + 78 x^{10} - 75 x^{9} - 36 x^{8} + 36 x^{7} + 81 x^{6} - 81 x^{5} - 81 x^{4} + 81 x^{3} + 81 x^{2} + 81 x - 81 \)

Invariants

Base field: $\Q_{3}$
Degree $d$ : $12$
Ramification exponent $e$ : $3$
Residue field degree $f$ : $4$
Discriminant exponent $c$ : $12$
Discriminant root field: $\Q_{3}$
Root number: $1$
$|\Aut(K/\Q_{ 3 })|$: $1$
This field is not Galois over $\Q_{3}$.

Intermediate fields

$\Q_{3}(\sqrt{*})$, 3.4.0.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

Unramified subfield:3.4.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{4} - x + 2 \)
Relative Eisenstein polynomial:$ x^{3} + \left(3 t^{3} - 3 t^{2} + 3 t\right) x^{2} + \left(-3 t^{3} - 3 t^{2} - 3 t\right) x - 3 t^{3} - 3 t^{2} \in\Q_{3}(t)[x]$

Invariants of the Galois closure

Galois group:12T173
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:$x^{12} - 12 x^{10} - 9 x^{9} + 54 x^{8} + 81 x^{7} - 101 x^{6} - 243 x^{5} + 39 x^{4} + 343 x^{3} + 63 x^{2} - 300 x - 188$