Properties

Label 2.12.35.485
Base \(\Q_{2}\)
Degree \(12\)
e \(12\)
f \(1\)
c \(35\)
Galois group $C_2^4.(C_4\times S_3)$ (as 12T153)

Related objects

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Defining polynomial

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

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, 2.6.11.8

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

Invariants of the Galois closure

Galois group:$C_2^4.(C_4\times S_3)$ (as 12T153)
Inertia group:12T99
Unramified degree:$2$
Tame degree:$3$
Wild slopes:[8/3, 8/3, 3, 23/6, 23/6, 4]
Galois mean slope:$361/96$
Galois splitting model:$x^{12} + 12 x^{10} + 54 x^{8} + 100 x^{6} + 240 x^{4} - 900 x^{2} + 250$