Properties

Label 2.8.14.1
Base \(\Q_{2}\)
Degree \(8\)
e \(8\)
f \(1\)
c \(14\)
Galois group $A_4\times C_2$ (as 8T13)

Related objects

Learn more about

Defining polynomial

\( x^{8} + 2 x^{7} + 6 \)

Invariants

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

Intermediate fields

$\Q_{2}(\sqrt{-*})$, 2.4.6.7

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:\( x^{8} + 2 x^{7} + 6 \)

Invariants of the Galois closure

Galois group:$C_2\times A_4$ (as 8T13)
Inertia group:$C_2^3$
Unramified degree:$3$
Tame degree:$1$
Wild slopes:[2, 2, 2]
Galois mean slope:$7/4$
Galois splitting model:$x^{8} - 2 x^{7} - 2 x^{6} + 2 x^{5} + 8 x^{4} + 2 x^{3} - 2 x^{2} - 2 x + 1$