Properties

Label 2.12.12.33
Base \(\Q_{2}\)
Degree \(12\)
e \(4\)
f \(3\)
c \(12\)
Galois group $C_3\times S_4$ (as 12T45)

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Defining polynomial

\( x^{12} + 6 x^{11} - 4 x^{9} - 2 x^{8} + 8 x^{7} + 8 x^{6} - 4 x^{5} + 8 x^{3} + 8 x^{2} + 8 \)

Invariants

Base field: $\Q_{2}$
Degree $d$ : $12$
Ramification exponent $e$ : $4$
Residue field degree $f$ : $3$
Discriminant exponent $c$ : $12$
Discriminant root field: $\Q_{2}(\sqrt{*})$
Root number: $i$
$|\Aut(K/\Q_{ 2 })|$: $3$
This field is not Galois over $\Q_{2}$.

Intermediate fields

2.3.0.1, 2.4.4.5

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

Unramified/totally ramified tower

Unramified subfield:2.3.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{3} - x + 1 \)
Relative Eisenstein polynomial:$ x^{4} + 2 x^{3} + \left(2 t^{2} + 2 t\right) x^{2} + \left(2 t^{2} + 2 t\right) x + 2 t + 2 \in\Q_{2}(t)[x]$

Invariants of the Galois closure

Galois group:$C_3\times S_4$ (as 12T45)
Inertia group:Intransitive group isomorphic to $A_4$
Unramified degree:$6$
Tame degree:$3$
Wild slopes:[4/3, 4/3]
Galois mean slope:$7/6$
Galois splitting model:\( x^{12} - 4 x^{9} + 54 x^{8} - 54 x^{7} + 122 x^{6} - 72 x^{5} + 93 x^{4} - 48 x^{3} + 18 x^{2} + 1 \)