Properties

Label 2.8.12.1
Base \(\Q_{2}\)
Degree \(8\)
e \(2\)
f \(4\)
c \(12\)
Galois group $C_4\times C_2$ (as 8T2)

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Defining polynomial

\(x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216\) 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: $\Q_{2}$
Root number: $-1$
$\card{ \Gal(K/\Q_{ 2 }) }$: $8$
This field is Galois and abelian over $\Q_{2}.$
Visible slopes:$[3]$

Intermediate fields

$\Q_{2}(\sqrt{5})$, $\Q_{2}(\sqrt{2})$, $\Q_{2}(\sqrt{2\cdot 5})$, 2.4.0.1, 2.4.6.1, 2.4.6.3

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: \( x^{2} + \left(4 t^{3} + 4 t^{2}\right) x + 12 t^{3} + 12 t^{2} + 4 t + 2 \) $\ \in\Q_{2}(t)[x]$ 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:$C_2\times C_4$ (as 8T2)
Inertia group:Intransitive group isomorphic to $C_2$
Wild inertia group:$C_2$
Unramified degree:$4$
Tame degree:$1$
Wild slopes:$[3]$
Galois mean slope:$3/2$
Galois splitting model:$x^{8} + 2 x^{6} + 4 x^{4} + 8 x^{2} + 16$