Properties

Label 2.12.18.31
Base \(\Q_{2}\)
Degree \(12\)
e \(4\)
f \(3\)
c \(18\)
Galois group $A_4\times C_2$ (as 12T6)

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Defining polynomial

\(x^{12} + 14 x^{11} + 4 x^{10} + 12 x^{9} + 8 x^{8} + 4 x^{7} + 12 x^{6} - 8 x^{5} + 4 x^{4} + 8 x^{3} + 16 x^{2} + 16 x - 8\)  Toggle raw display

Invariants

Base field: $\Q_{2}$
Degree $d$: $12$
Ramification exponent $e$: $4$
Residue field degree $f$: $3$
Discriminant exponent $c$: $18$
Discriminant root field: $\Q_{2}$
Root number: $1$
$|\Aut(K/\Q_{ 2 })|$: $4$
This field is not Galois over $\Q_{2}.$

Intermediate fields

2.3.0.1, 2.6.6.1, 2.6.6.4 x2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

Unramified subfield:2.3.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{3} - x + 1 \)  Toggle raw display
Relative Eisenstein polynomial:\( x^{4} + \left(4 t^{2} - 2 t - 2\right) x^{3} + \left(4 t^{2} + 2 t + 4\right) x^{2} + 4 x - 2 t^{2} - 2 t \)$\ \in\Q_{2}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_2\times A_4$ (as 12T6)
Inertia group:Intransitive group isomorphic to $C_2^3$
Unramified degree:$3$
Tame degree:$1$
Wild slopes:[2, 2, 2]
Galois mean slope:$7/4$
Galois splitting model:$x^{12} - 6 x^{11} + 12 x^{10} + 4 x^{9} - 82 x^{8} + 240 x^{7} - 424 x^{6} + 528 x^{5} - 480 x^{4} + 320 x^{3} - 152 x^{2} + 48 x - 8$  Toggle raw display