Properties

Label 3.12.14.13
Base \(\Q_{3}\)
Degree \(12\)
e \(6\)
f \(2\)
c \(14\)
Galois group $C_3\times C_3:S_3.C_2$ (as 12T73)

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Defining polynomial

\( x^{12} + 9 x^{11} - 9 x^{10} - 3 x^{9} - 6 x^{8} + 9 x^{7} - 6 x^{6} + 9 x^{4} + 9 x^{3} - 9 \)

Invariants

Base field: $\Q_{3}$
Degree $d$ : $12$
Ramification exponent $e$ : $6$
Residue field degree $f$ : $2$
Discriminant exponent $c$ : $14$
Discriminant root field: $\Q_{3}(\sqrt{*})$
Root number: $-1$
$|\Aut(K/\Q_{ 3 })|$: $3$
This field is not Galois over $\Q_{3}$.

Intermediate fields

$\Q_{3}(\sqrt{*})$, 3.4.2.2

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} + \left(3 t + 3\right) x^{5} + \left(3 t + 3\right) x^{4} + 3 x^{3} + \left(-3 t + 3\right) x^{2} + 3 t \in\Q_{3}(t)[x]$

Invariants of the Galois closure

Galois group:$C_3\times C_3:S_3.C_2$ (as 12T73)
Inertia group:Intransitive group isomorphic to $C_3:S_3$
Unramified degree:$6$
Tame degree:$2$
Wild slopes:[3/2, 3/2]
Galois mean slope:$25/18$
Galois splitting model:$x^{12} + 84 x^{10} - 196 x^{9} + 2646 x^{8} - 12348 x^{7} + 54446 x^{6} - 259308 x^{5} + 925365 x^{4} - 2600528 x^{3} + 7674282 x^{2} - 16492812 x + 19131511$