Properties

Label 2.12.16.13
Base \(\Q_{2}\)
Degree \(12\)
e \(6\)
f \(2\)
c \(16\)
Galois group $D_6$

Related objects

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Defining polynomial

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

Invariants

Base field: $\Q_{2}$
Degree $d$ : $12$
Ramification exponent $e$ : $6$
Residue field degree $f$ : $2$
Discriminant exponent $c$ : $16$
Discriminant root field: $\Q_{2}$
Root number: $-i$
$|\Gal(K/\Q_{ 2 })|$: $12$
This field is Galois over $\Q_{2}$.

Intermediate fields

$\Q_{2}(\sqrt{*})$, $\Q_{2}(\sqrt{-1})$, $\Q_{2}(\sqrt{-*})$, 2.3.2.1 x3, 2.4.4.1, 2.6.4.1, 2.6.8.1 x3, 2.6.8.3 x3

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

Invariants of the Galois closure

Galois group:$D_6$
Inertia group:Intransitive group isomorphic to $C_6$
Unramified degree:$2$
Tame degree:$3$
Wild slopes:[2]
Galois mean slope:$4/3$
Global splitting model:\( x^{12} - 3 x^{10} + 2 x^{8} + x^{6} + 2 x^{4} - 3 x^{2} + 1 \)