Properties

Label 2.12.14.2
Base \(\Q_{2}\)
Degree \(12\)
e \(12\)
f \(1\)
c \(14\)
Galois group $A_4:C_4$ (as 12T27)

Related objects

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Defining polynomial

\( x^{12} + 2 x^{4} + 2 x^{3} + 2 \)

Invariants

Base field: $\Q_{2}$
Degree $d$ : $12$
Ramification exponent $e$ : $12$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $14$
Discriminant root field: $\Q_{2}(\sqrt{*})$
Root number: $i$
$|\Aut(K/\Q_{ 2 })|$: $2$
This field is not Galois over $\Q_{2}$.

Intermediate fields

2.3.2.1, 2.6.6.8

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{2}$
Relative Eisenstein polynomial:\( x^{12} + 2 x^{4} + 2 x^{3} + 2 \)

Invariants of the Galois closure

Galois group:$A_4:C_4$ (as 12T27)
Inertia group:$A_4$
Unramified degree:$4$
Tame degree:$3$
Wild slopes:[4/3, 4/3]
Galois mean slope:$7/6$
Galois splitting model:\( x^{12} - 2 x^{11} + 2 x^{10} + 37 x^{8} + 32 x^{7} - 124 x^{6} + 14 x^{5} + 311 x^{4} - 488 x^{3} + 360 x^{2} - 144 x + 27 \)