Properties

Label 2.12.12.31
Base \(\Q_{2}\)
Degree \(12\)
e \(4\)
f \(3\)
c \(12\)
Galois group 12T205

Related objects

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Defining polynomial

\( x^{12} + 4 x^{11} - 6 x^{10} + 8 x^{9} - 4 x^{8} + 8 x^{7} - 4 x^{6} + 4 x^{5} - 4 x^{4} + 8 x + 8 \)

Invariants

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

Intermediate fields

2.3.0.1

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

Unramified/totally ramified tower

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

Invariants of the Galois closure

Galois group:12T205
Inertia group:Intransitive group isomorphic to $C_2^6:C_3$
Unramified degree:$6$
Tame degree:$3$
Wild slopes:[4/3, 4/3, 4/3, 4/3, 4/3, 4/3]
Galois mean slope:$127/96$
Global splitting model:\( x^{12} - 4 x^{11} - 52 x^{10} + 156 x^{9} + 1224 x^{8} - 1876 x^{7} - 14588 x^{6} + 3738 x^{5} + 74477 x^{4} + 37070 x^{3} - 84064 x^{2} + 22966 x + 102679 \)