Properties

Label 2.12.16.12
Base \(\Q_{2}\)
Degree \(12\)
e \(6\)
f \(2\)
c \(16\)
Galois group 12T159

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Defining polynomial

\( x^{12} + 18 x^{10} + 171 x^{8} - 404 x^{6} - 281 x^{4} - 286 x^{2} + 461 \)

Invariants

Base field: $\Q_{2}$
Degree $d$ : $12$
Ramification exponent $e$ : $6$
Residue field degree $f$ : $2$
Discriminant exponent $c$ : $16$
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} + 4 x^{5} + 2 t x^{4} + \left(-2 t + 2\right) x^{3} + 4 t x^{2} + \left(4 t + 4\right) x - 2 t + 4 \in\Q_{2}(t)[x]$

Invariants of the Galois closure

Galois group:12T159
Inertia group:Intransitive group isomorphic to $C_2\times C_2^4:C_3$
Unramified degree:$6$
Tame degree:$3$
Wild slopes:[4/3, 4/3, 4/3, 4/3, 2]
Galois mean slope:$79/48$
Galois splitting model:\( x^{12} - 33 x^{10} - 62 x^{9} + 324 x^{8} + 882 x^{7} - 583 x^{6} - 3744 x^{5} - 2940 x^{4} + 2318 x^{3} + 4401 x^{2} + 1986 x + 211 \)