Properties

Label 2.12.12.29
Base \(\Q_{2}\)
Degree \(12\)
e \(6\)
f \(2\)
c \(12\)
Galois group 12T159

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Defining polynomial

\( x^{12} + 6 x^{10} + 51 x^{8} - 252 x^{6} - 393 x^{4} - 234 x^{2} - 203 \)

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: $i$
$|\Aut(K/\Q_{ 2 })|$: $2$
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} + 2 t x^{5} + \left(2 t + 2\right) x^{4} + \left(2 t + 2\right) x^{3} + 2 t x^{2} + \left(2 t + 2\right) x + 2 t + 2 \in\Q_{2}(t)[x]$

Invariants of the Galois closure

Galois group:12T159
Inertia group:Intransitive group isomorphic to $C_2^4:C_3$
Unramified degree:$12$
Tame degree:$3$
Wild slopes:[4/3, 4/3, 4/3, 4/3]
Galois mean slope:$31/24$
Galois splitting model:\( x^{12} - 27 x^{10} - 86 x^{9} + 54 x^{8} + 1044 x^{7} + 3423 x^{6} + 6318 x^{5} + 7200 x^{4} + 4966 x^{3} + 2079 x^{2} + 648 x + 151 \)