Properties

Label 2.12.12.20
Base \(\Q_{2}\)
Degree \(12\)
e \(2\)
f \(6\)
c \(12\)
Galois group 12T87

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Defining polynomial

\( x^{12} - 18 x^{10} - 49 x^{8} - 52 x^{6} + 39 x^{4} + 6 x^{2} + 9 \)

Invariants

Base field: $\Q_{2}$
Degree $d$ : $12$
Ramification exponent $e$ : $2$
Residue field degree $f$ : $6$
Discriminant exponent $c$ : $12$
Discriminant root field: $\Q_{2}$
Root number: $-i$
$|\Aut(K/\Q_{ 2 })|$: $2$
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^{2}\right) x + 2 t^{3} + 4 t^{2} + 4 t + 4 \in\Q_{2}(t)[x]$

Invariants of the Galois closure

Galois group:12T87
Inertia group:Intransitive group isomorphic to $C_2^5$
Unramified degree:$6$
Tame degree:$1$
Wild slopes:[2, 2, 2, 2, 2]
Galois mean slope:$31/16$
Global splitting model:\( x^{12} + 6 x^{10} - 11 x^{8} - 6 x^{6} + 15 x^{4} - 7 x^{2} + 1 \)