Properties

Label 2.12.12.27
Base \(\Q_{2}\)
Degree \(12\)
e \(6\)
f \(2\)
c \(12\)
Galois group 12T30

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Defining polynomial

\( x^{12} - 18 x^{10} + 171 x^{8} + 116 x^{6} - 313 x^{4} + 190 x^{2} + 877 \)

Invariants

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

Intermediate fields

$\Q_{2}(\sqrt{*})$, 2.3.2.1 x3, 2.6.4.1

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

Invariants of the Galois closure

Galois group:12T30
Inertia group:Intransitive group isomorphic to $A_4$
Unramified degree:$4$
Tame degree:$3$
Wild slopes:[4/3, 4/3]
Galois mean slope:$7/6$
Global splitting model:\( x^{12} - 2 x^{11} - 13 x^{10} + 6 x^{9} + 10 x^{8} + 56 x^{7} + 47 x^{6} - 26 x^{5} - 110 x^{4} - 124 x^{3} - 63 x^{2} - 14 x - 1 \)