Properties

Label 2.12.16.11
Base \(\Q_{2}\)
Degree \(12\)
e \(6\)
f \(2\)
c \(16\)
Galois group $C_3\times (C_3 : C_4)$

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Defining polynomial

\( x^{12} + 20 x^{10} - 44 x^{8} - 4 x^{6} - 16 x^{4} - 48 \)

Invariants

Base field: $\Q_{2}$
Degree $d$ : $12$
Ramification exponent $e$ : $6$
Residue field degree $f$ : $2$
Discriminant exponent $c$ : $16$
Discriminant root field: $\Q_{2}(\sqrt{*})$
Root number: $i$
$|\Aut(K/\Q_{ 2 })|$: $6$
This field is not Galois over $\Q_{2}$.

Intermediate fields

$\Q_{2}(\sqrt{*})$, 2.4.4.2, 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:$\Q_{2}(\sqrt{*})$ $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{2} - x + 1 \)
Relative Eisenstein polynomial:$ x^{6} + \left(4 t - 2\right) x^{5} - 2 x^{4} + \left(-2 t - 2\right) x^{3} + 4 t x^{2} + 4 t x + 2 t + 4 \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 $C_6$
Unramified degree:$6$
Tame degree:$3$
Wild slopes:[2]
Galois mean slope:$4/3$
Global splitting model:\( x^{12} - 39 x^{10} + 12 x^{9} + 486 x^{8} - 156 x^{7} - 2175 x^{6} - 204 x^{5} + 3738 x^{4} + 1616 x^{3} - 2031 x^{2} - 1488 x - 199 \)