Properties

Label 2.12.12.34
Base \(\Q_{2}\)
Degree \(12\)
e \(12\)
f \(1\)
c \(12\)
Galois group 12T254

Related objects

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Defining polynomial

\( x^{12} + 2 x^{11} + 2 x^{9} + 2 x^{7} + 2 x^{5} + 2 x^{3} + 2 x + 2 \)

Invariants

Base field: $\Q_{2}$
Degree $d$ : $12$
Ramification exponent $e$ : $12$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $12$
Discriminant root field: $\Q_{2}(\sqrt{*})$
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:\( x^{12} + 2 x^{11} + 2 x^{9} + 2 x^{7} + 2 x^{5} + 2 x^{3} + 2 x + 2 \)

Invariants of the Galois closure

Galois group:12T254
Inertia group:12T166
Unramified degree:$6$
Tame degree:$9$
Wild slopes:[10/9, 10/9, 10/9, 10/9, 10/9, 10/9]
Galois mean slope:$319/288$
Global splitting model:\( x^{12} - 18 x^{10} - 186 x^{9} + 1179 x^{8} - 1818 x^{7} + 2874 x^{6} - 36360 x^{5} + 192951 x^{4} - 519944 x^{3} + 835722 x^{2} - 786834 x + 359163 \)