Properties

Label 3.12.12.27
Base \(\Q_{3}\)
Degree \(12\)
e \(6\)
f \(2\)
c \(12\)
Galois group $PSU(3,2)$ (as 12T47)

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Defining polynomial

\( x^{12} - 12 x^{11} + 3 x^{10} - 9 x^{9} + 3 x^{8} + 3 x^{7} + 6 x^{6} + 9 x^{3} + 9 x + 9 \)

Invariants

Base field: $\Q_{3}$
Degree $d$ : $12$
Ramification exponent $e$ : $6$
Residue field degree $f$ : $2$
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

$\Q_{3}(\sqrt{*})$, $\Q_{3}(\sqrt{3})$, $\Q_{3}(\sqrt{3*})$, 3.4.2.1

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{3}(\sqrt{*})$ $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{2} - x + 2 \)
Relative Eisenstein polynomial:$ x^{6} + 3 x^{5} + 3 x^{4} + \left(3 t + 3\right) x^{3} + \left(3 t + 3\right) x^{2} + 3 t x + 3 \in\Q_{3}(t)[x]$

Invariants of the Galois closure

Galois group:$PSU(3,2)$ (as 12T47)
Inertia group:Intransitive group isomorphic to $C_3:S_3.C_2$
Unramified degree:$2$
Tame degree:$4$
Wild slopes:[5/4, 5/4]
Galois mean slope:$43/36$
Galois splitting model:$x^{12} - 8 x^{9} - 18 x^{8} - 24 x^{7} - 28 x^{6} - 24 x^{5} + 30 x^{4} + 32 x^{3} + 48 x^{2} - 32$