Properties

Label 2.12.12.5
Base \(\Q_{2}\)
Degree \(12\)
e \(2\)
f \(6\)
c \(12\)
Galois group $D_4\times A_4$ (as 12T51)

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Defining polynomial

\( x^{12} + 52 x^{10} - 11 x^{8} - 8 x^{6} - 45 x^{4} - 44 x^{2} - 9 \)

Invariants

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

Intermediate fields

$\Q_{2}(\sqrt{*})$, 2.3.0.1, 2.6.0.1

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

Unramified/totally ramified tower

Unramified subfield:2.6.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{6} - x + 1 \)
Relative Eisenstein polynomial:$ x^{2} + \left(2 t^{4} + 2 t^{3} + 2 t + 2\right) x + 2 t^{4} + 2 \in\Q_{2}(t)[x]$

Invariants of the Galois closure

Galois group:$D_4\times A_4$ (as 12T51)
Inertia group:Intransitive group isomorphic to $C_2^4$
Unramified degree:$6$
Tame degree:$1$
Wild slopes:[2, 2, 2, 2]
Galois mean slope:$15/8$
Galois splitting model:\( x^{12} - 4 x^{10} - 2 x^{8} + 9 x^{6} + 2 x^{4} - 4 x^{2} - 1 \)