Properties

Label 2.12.12.28
Base \(\Q_{2}\)
Degree \(12\)
e \(6\)
f \(2\)
c \(12\)
Galois group $S_4$

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Defining polynomial

\( x^{12} - x^{10} + 2 x^{8} - x^{6} - 2 x^{4} + 3 x^{2} + 1 \)

Invariants

Base field: $\Q_{2}$
Degree $d$ : $12$
Ramification exponent $e$ : $6$
Residue field degree $f$ : $2$
Discriminant exponent $c$ : $12$
Discriminant root field: $\Q_{2}$
Root number: $-i$
$|\Aut(K/\Q_{ 2 })|$: $4$
This field is not Galois over $\Q_{2}$.

Intermediate fields

$\Q_{2}(\sqrt{*})$, 2.3.2.1 x3, 2.6.4.1, 2.6.6.7, 2.6.6.8

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{2}(\sqrt{*})$ $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{2} - x + 1 \)
Relative Eisenstein polynomial:$ x^{6} + 2 x^{4} + \left(2 t + 2\right) x^{3} + \left(2 t + 2\right) x + 2 \in\Q_{2}(t)[x]$

Invariants of the Galois closure

Galois group:$S_4$
Inertia group:Intransitive group isomorphic to $A_4$
Unramified degree:$2$
Tame degree:$3$
Wild slopes:[4/3, 4/3]
Galois mean slope:$7/6$
Global splitting model:\( x^{12} - 4 x^{11} + x^{10} + 14 x^{9} - 14 x^{8} - 10 x^{7} + 25 x^{6} - 10 x^{5} - 14 x^{4} + 14 x^{3} + x^{2} - 4 x + 1 \)