Properties

Label 2.8.14.14
Base \(\Q_{2}\)
Degree \(8\)
e \(8\)
f \(1\)
c \(14\)
Galois group $C_2 \wr S_4$ (as 8T44)

Related objects

Learn more about

Defining polynomial

\(x^{8} + 4 x^{2} + 12\)  Toggle raw display

Invariants

Base field: $\Q_{2}$
Degree $d$: $8$
Ramification exponent $e$: $8$
Residue field degree $f$: $1$
Discriminant exponent $c$: $14$
Discriminant root field: $\Q_{2}(\sqrt{-5})$
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:\( x^{8} - 6 x^{7} + 18 x^{6} - 40 x^{5} + 54 x^{4} - 36 x^{3} + 26 x^{2} - 12 x + 18 \)  Toggle raw display

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:$x^{8} + 4 x^{2} + 12$