Properties

Label 2.8.12.11
Base \(\Q_{2}\)
Degree \(8\)
e \(2\)
f \(4\)
c \(12\)
Galois group $((C_8 : C_2):C_2):C_2$ (as 8T27)

Related objects

Downloads

Learn more

Defining polynomial

\(x^{8} + 16 x^{7} + 92 x^{6} + 352 x^{5} + 1264 x^{4} + 2304 x^{3} + 6544 x^{2} - 256 x + 12080\) Copy content Toggle raw display

Invariants

Base field: $\Q_{2}$
Degree $d$: $8$
Ramification exponent $e$: $2$
Residue field degree $f$: $4$
Discriminant exponent $c$: $12$
Discriminant root field: $\Q_{2}(\sqrt{-5})$
Root number: $-i$
$\card{ \Aut(K/\Q_{ 2 }) }$: $2$
This field is not Galois over $\Q_{2}.$
Visible slopes:$[3]$

Intermediate fields

$\Q_{2}(\sqrt{5})$, 2.4.0.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

Unramified subfield:2.4.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{4} + x + 1 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{2} + \left(4 t + 4\right) x + 4 t^{3} + 12 t^{2} + 4 t + 2 \) $\ \in\Q_{2}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z + 1$
Associated inertia:$1$
Indices of inseparability:$[2, 0]$

Invariants of the Galois closure

Galois group:$C_2\wr C_4$ (as 8T27)
Inertia group:Intransitive group isomorphic to $C_2^4$
Wild inertia group:$C_2^4$
Unramified degree:$4$
Tame degree:$1$
Wild slopes:$[2, 2, 2, 3]$
Galois mean slope:$19/8$
Galois splitting model:$x^{8} - 10 x^{6} + 20 x^{4} + 40 x^{2} - 80$