Properties

Label 11.8.6.3
Base \(\Q_{11}\)
Degree \(8\)
e \(4\)
f \(2\)
c \(6\)
Galois group $C_8:C_2$ (as 8T7)

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Defining polynomial

\(x^{8} - 165 x^{4} - 4356\) Copy content Toggle raw display

Invariants

Base field: $\Q_{11}$
Degree $d$: $8$
Ramification exponent $e$: $4$
Residue field degree $f$: $2$
Discriminant exponent $c$: $6$
Discriminant root field: $\Q_{11}(\sqrt{2})$
Root number: $-1$
$\card{ \Aut(K/\Q_{ 11 }) }$: $4$
This field is not Galois over $\Q_{11}.$
Visible slopes:None

Intermediate fields

$\Q_{11}(\sqrt{2})$, 11.4.2.2

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{11}(\sqrt{2})$ $\cong \Q_{11}(t)$ where $t$ is a root of \( x^{2} + 7 x + 2 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{4} + 33 t + 33 \) $\ \in\Q_{11}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{3} + 4z^{2} + 6z + 4$
Associated inertia:$1$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$\OD_{16}$ (as 8T7)
Inertia group:Intransitive group isomorphic to $C_4$
Wild inertia group:$C_1$
Unramified degree:$4$
Tame degree:$4$
Wild slopes:None
Galois mean slope:$3/4$
Galois splitting model:$x^{8} - 4 x^{7} + 7 x^{6} - 7 x^{5} - 371 x^{4} + 749 x^{3} - 366 x^{2} - 9 x + 81$