Properties

Label 2.12.35.340
Base \(\Q_{2}\)
Degree \(12\)
e \(12\)
f \(1\)
c \(35\)
Galois group 12T225

Related objects

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Defining polynomial

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

Invariants

Base field: $\Q_{ 2 }$
Degree $d$ : $12$
Ramification exponent $e$ : $12$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $35$
Discriminant root field: $\Q_{2}(\sqrt{2*})$
Root number: $-i$
$|\Aut(K/\Q_{ 2 })|$: $2$
This field is not Galois over $\Q_{2}$.

Intermediate fields

2.3.2.1, 2.6.11.4

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

Unramified/totally ramified tower

Unramified subfield:\( t + 1 \)
Relative Eisenstein polynomial:\( y^{12} - 4 y^{10} - 2 y^{8} + 4 y^{6} + 4 y^{4} - 4 y^{2} - 6 \)

Invariants of the Galois closure

Galois group:12T225
Inertia group:12T189
Unramified degree:$2$
Tame degree:$3$
Wild slopes:[4/3, 4/3, 8/3, 8/3, 3, 23/6, 23/6, 4]
Galois Mean Slope:$1447/384$
Galois Splitting Model:\( x^{12} + 24 x^{10} + 100 x^{8} - 792 x^{6} - 352 x^{4} + 3388 x^{2} - 2662 \)