Properties

Label 2.12.14.3
Base \(\Q_{2}\)
Degree \(12\)
e \(12\)
f \(1\)
c \(14\)
Galois group $A_4\wr C_2$ (as 12T128)

Related objects

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Defining polynomial

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

Invariants

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

Intermediate fields

2.3.2.1

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

Unramified/totally ramified tower

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

Invariants of the Galois closure

Galois group:$A_4\wr C_2$ (as 12T128)
Inertia group:$C_2^4:C_3$
Unramified degree:$6$
Tame degree:$3$
Wild slopes:[4/3, 4/3, 4/3, 4/3]
Galois mean slope:$31/24$
Galois splitting model:\( x^{12} - 4 x^{11} - 6 x^{10} + 48 x^{9} - 26 x^{8} - 192 x^{7} + 216 x^{6} + 400 x^{5} - 416 x^{4} - 394 x^{3} + 314 x^{2} + 120 x - 106 \)