Properties

Label 2.12.12.30
Base \(\Q_{2}\)
Degree \(12\)
e \(6\)
f \(2\)
c \(12\)
Galois group $A_4\wr C_2$ (as 12T126)

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Defining polynomial

\( x^{12} - 6 x^{10} + 15 x^{8} - 52 x^{6} + 111 x^{4} - 102 x^{2} - 991 \)

Invariants

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

Intermediate fields

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

Invariants of the Galois closure

Galois group:$A_4\wr C_2$ (as 12T126)
Inertia group:Intransitive group isomorphic to $C_2^4:C_3$
Unramified degree:$6$
Tame degree:$3$
Wild slopes:[4/3, 4/3, 4/3, 4/3]
Galois mean slope:$31/24$
Galois splitting model:\( x^{12} + 2 x^{11} - x^{8} - 2 x^{7} - 4 x^{6} - 2 x^{5} - x^{4} + 2 x + 1 \)