Properties

Label 2.12.24.62
Base \(\Q_{2}\)
Degree \(12\)
e \(4\)
f \(3\)
c \(24\)
Galois group $C_2^5.(C_2\times C_6)$ (as 12T134)

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Defining polynomial

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

Invariants

Base field: $\Q_{2}$
Degree $d$: $12$
Ramification exponent $e$: $4$
Residue field degree $f$: $3$
Discriminant exponent $c$: $24$
Discriminant root field: $\Q_{2}(\sqrt{*})$
Root number: $-1$
$|\Aut(K/\Q_{ 2 })|$: $2$
This field is not Galois over $\Q_{2}.$

Intermediate fields

$\Q_{2}(\sqrt{-1})$, 2.3.0.1, 2.6.6.3

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 \( x^{3} - x + 1 \)
Relative Eisenstein polynomial:$ x^{4} + 4 t x^{3} + \left(-2 t^{2} - 2 t\right) x^{2} + 4 t x + 4 t^{2} - 2 t \in\Q_{2}(t)[x]$

Invariants of the Galois closure

Galois group:$C_2^5.(C_2\times C_6)$ (as 12T134)
Inertia group:Intransitive group isomorphic to $C_2\times C_2^2\wr C_2$
Unramified degree:$6$
Tame degree:$1$
Wild slopes:[2, 2, 2, 3, 3, 3]
Galois mean slope:$91/32$
Galois splitting model:$x^{12} + 12 x^{10} + 11 x^{8} - 134 x^{6} + 156 x^{4} - 18 x^{2} - 27$