Properties

Label 2.12.12.25
Base \(\Q_{2}\)
Degree \(12\)
e \(2\)
f \(6\)
c \(12\)
Galois group $C_{12}$ (as 12T1)

Related objects

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Defining polynomial

\( x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749 \)

Invariants

Base field: $\Q_{2}$
Degree $d$ : $12$
Ramification exponent $e$ : $2$
Residue field degree $f$ : $6$
Discriminant exponent $c$ : $12$
Discriminant root field: $\Q_{2}(\sqrt{*})$
Root number: $i$
$|\Gal(K/\Q_{ 2 })|$: $12$
This field is Galois and abelian over $\Q_{2}$.

Intermediate fields

$\Q_{2}(\sqrt{*})$, 2.3.0.1, 2.4.4.2, 2.6.0.1

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

Unramified/totally ramified tower

Unramified subfield:2.6.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{6} - x + 1 \)
Relative Eisenstein polynomial:$ x^{2} + \left(-2 t^{2} + 4 t + 4\right) x + 2 t^{4} + 4 \in\Q_{2}(t)[x]$

Invariants of the Galois closure

Galois group:$C_{12}$ (as 12T1)
Inertia group:Intransitive group isomorphic to $C_2$
Unramified degree:$6$
Tame degree:$1$
Wild slopes:[2]
Galois mean slope:$1$
Galois splitting model:\( x^{12} - 4 x^{11} - 17 x^{10} + 74 x^{9} + 74 x^{8} - 412 x^{7} - 23 x^{6} + 734 x^{5} - 175 x^{4} - 324 x^{3} + 90 x^{2} + 22 x + 1 \)