Properties

Label 2.12.8.1
Base \(\Q_{2}\)
Degree \(12\)
e \(3\)
f \(4\)
c \(8\)
Galois group $C_3 : C_4$

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Defining polynomial

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

Invariants

Base field: $\Q_{2}$
Degree $d$ : $12$
Ramification exponent $e$ : $3$
Residue field degree $f$ : $4$
Discriminant exponent $c$ : $8$
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, 2.4.0.1, 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:2.4.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{4} - x + 1 \)
Relative Eisenstein polynomial:$ x^{3} + \left(2 t^{3} + 2 t^{2} + 2 t + 2\right) x^{2} + 2 \in\Q_{2}(t)[x]$

Invariants of the Galois closure

Galois group:$C_3 : C_4$
Inertia group:Intransitive group isomorphic to $C_3$
Unramified degree:$4$
Tame degree:$3$
Wild slopes:None
Galois mean slope:$2/3$
Global splitting model:\( x^{12} - 3 x^{11} - 888 x^{10} + 2660 x^{9} + 308151 x^{8} - 878220 x^{7} - 53209587 x^{6} + 132388596 x^{5} + 4800412971 x^{4} - 8975649010 x^{3} - 213443493450 x^{2} + 211625342025 x + 3644274033955 \)