Properties

Label 2.12.8.2
Base \(\Q_{2}\)
Degree \(12\)
e \(3\)
f \(4\)
c \(8\)
Galois group $C_3\times (C_3 : C_4)$

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Defining polynomial

\( x^{12} - 52 x^{10} + 20 x^{8} - 60 x^{6} - 32 x^{4} - 16 x^{2} - 48 \)

Invariants

Base field: $\Q_{2}$
Degree $d$ : $12$
Ramification exponent $e$ : $3$
Residue field degree $f$ : $4$
Discriminant exponent $c$ : $8$
Discriminant root field: $\Q_{2}(\sqrt{*})$
Root number: $i$
$|\Aut(K/\Q_{ 2 })|$: $6$
This field is not Galois over $\Q_{2}$.

Intermediate fields

$\Q_{2}(\sqrt{*})$, 2.4.0.1, 2.6.4.2

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

Unramified/totally ramified tower

Unramified subfield:2.4.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{4} - x + 1 \)
Relative Eisenstein polynomial:$ x^{3} + 2 x^{2} + \left(2 t^{2} + 2 t + 2\right) x + 2 t^{2} + 2 t \in\Q_{2}(t)[x]$

Invariants of the Galois closure

Galois group:$C_3\times (C_3 : C_4)$
Inertia group:Intransitive group isomorphic to $C_3$
Unramified degree:$12$
Tame degree:$3$
Wild slopes:None
Galois mean slope:$2/3$
Global splitting model:\( x^{12} - 24 x^{10} - 8 x^{9} + 216 x^{8} + 144 x^{7} - 880 x^{6} - 864 x^{5} + 1488 x^{4} + 1976 x^{3} - 576 x^{2} - 1488 x - 464 \)