Properties

Label 2.12.16.18
Base \(\Q_{2}\)
Degree \(12\)
e \(6\)
f \(2\)
c \(16\)
Galois group $C_3 : C_4$ (as 12T5)

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Defining polynomial

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

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: $i$
$|\Gal(K/\Q_{ 2 })|$: $12$
This field is Galois over $\Q_{2}$.

Intermediate fields

$\Q_{2}(\sqrt{*})$, 2.3.2.1 x3, 2.4.4.2, 2.6.4.1

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} + \left(4 t - 2\right) x^{5} + 4 t x^{4} + \left(4 t - 2\right) x^{3} + \left(4 t + 2\right) x^{2} + \left(4 t + 4\right) x + 4 t + 2 \in\Q_{2}(t)[x]$

Invariants of the Galois closure

Galois group:$C_3:C_4$ (as 12T5)
Inertia group:Intransitive group isomorphic to $C_6$
Unramified degree:$2$
Tame degree:$3$
Wild slopes:[2]
Galois mean slope:$4/3$
Galois splitting model:$x^{12} - 6 x^{11} - 39 x^{10} + 126 x^{9} + 702 x^{8} - 354 x^{7} - 4589 x^{6} - 3012 x^{5} + 9954 x^{4} + 12828 x^{3} - 4167 x^{2} - 12600 x - 4819$