Properties

Label 2.8.14.10
Base \(\Q_{2}\)
Degree \(8\)
e \(8\)
f \(1\)
c \(14\)
Galois group $C_2 \wr S_4$ (as 8T44)

Related objects

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Defining polynomial

\( x^{8} + 4 x^{2} + 28 \)

Invariants

Base field: $\Q_{2}$
Degree $d$ : $8$
Ramification exponent $e$ : $8$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $14$
Discriminant root field: $\Q_{2}(\sqrt{-1})$
Root number: $-i$
$|\Aut(K/\Q_{ 2 })|$: $2$
This field is not Galois over $\Q_{2}$.

Intermediate fields

2.4.4.5

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:\( x^{8} - 2 x^{7} + 38 x^{6} + 92 x^{5} - 400 x^{4} - 4360 x^{3} + 14054 x^{2} + 83688 x + 429550 \)

Invariants of the Galois closure

Galois group:$C_2^3:S_4.C_2$ (as 8T44)
Inertia group:$C_2\wr A_4$
Unramified degree:$2$
Tame degree:$3$
Wild slopes:[4/3, 4/3, 2, 7/3, 7/3, 5/2]
Galois mean slope:$223/96$
Galois splitting model:$x^{8} + 4 x^{2} + 28$