Properties

Label 19.8.6.3
Base \(\Q_{19}\)
Degree \(8\)
e \(4\)
f \(2\)
c \(6\)
Galois group $C_8:C_2$ (as 8T7)

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Defining polynomial

\(x^{8} - 5510 x^{4} - 1650131\) Copy content Toggle raw display

Invariants

Base field: $\Q_{19}$
Degree $d$: $8$
Ramification exponent $e$: $4$
Residue field degree $f$: $2$
Discriminant exponent $c$: $6$
Discriminant root field: $\Q_{19}(\sqrt{2})$
Root number: $-1$
$\card{ \Aut(K/\Q_{ 19 }) }$: $4$
This field is not Galois over $\Q_{19}.$
Visible slopes:None

Intermediate fields

$\Q_{19}(\sqrt{2})$, 19.4.2.2

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{19}(\sqrt{2})$ $\cong \Q_{19}(t)$ where $t$ is a root of \( x^{2} + 18 x + 2 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{4} + 342 t + 323 \) $\ \in\Q_{19}(t)[x]$ 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:Not computed