Properties

Label 2.8.24.7
Base \(\Q_{2}\)
Degree \(8\)
e \(8\)
f \(1\)
c \(24\)
Galois group $C_4\times C_2$ (as 8T2)

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Defining polynomial

\(x^{8} + 8 x^{7} + 8 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 14\) Copy content Toggle raw display

Invariants

Base field: $\Q_{2}$
Degree $d$: $8$
Ramification exponent $e$: $8$
Residue field degree $f$: $1$
Discriminant exponent $c$: $24$
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:$[2, 3, 4]$

Intermediate fields

$\Q_{2}(\sqrt{-5})$, $\Q_{2}(\sqrt{-2\cdot 5})$, $\Q_{2}(\sqrt{2})$, 2.4.8.3, 2.4.11.4, 2.4.11.2

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: \( x^{8} + 8 x^{7} + 8 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 14 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z + 1$,$z^{2} + 1$,$z^{4} + 1$
Associated inertia:$1$,$1$,$1$
Indices of inseparability:$[17, 10, 4, 0]$

Invariants of the Galois closure

Galois group:$C_2\times C_4$ (as 8T2)
Inertia group:$C_2\times C_4$ (as 8T2)
Wild inertia group:$C_2\times C_4$
Unramified degree:$1$
Tame degree:$1$
Wild slopes:$[2, 3, 4]$
Galois mean slope:$3$
Galois splitting model:$x^{8} + 8 x^{6} + 20 x^{4} + 16 x^{2} + 1$