Properties

Label 19.8.4.2
Base \(\Q_{19}\)
Degree \(8\)
e \(2\)
f \(4\)
c \(4\)
Galois group $C_8$ (as 8T1)

Related objects

Downloads

Learn more

Defining polynomial

\(x^{8} + 722 x^{4} - 75449 x^{2} + 260642\) Copy content Toggle raw display

Invariants

Base field: $\Q_{19}$
Degree $d$: $8$
Ramification exponent $e$: $2$
Residue field degree $f$: $4$
Discriminant exponent $c$: $4$
Discriminant root field: $\Q_{19}(\sqrt{2})$
Root number: $1$
$\card{ \Gal(K/\Q_{ 19 }) }$: $8$
This field is Galois and abelian over $\Q_{19}.$
Visible slopes:None

Intermediate fields

$\Q_{19}(\sqrt{2})$, 19.4.0.1

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

Unramified/totally ramified tower

Unramified subfield:19.4.0.1 $\cong \Q_{19}(t)$ where $t$ is a root of \( x^{4} + 2 x^{2} + 11 x + 2 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{2} + 19 t \) $\ \in\Q_{19}(t)[x]$ Copy content Toggle raw display

Ramification polygon

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

Invariants of the Galois closure

Galois group:$C_8$ (as 8T1)
Inertia group:Intransitive group isomorphic to $C_2$
Wild inertia group:$C_1$
Unramified degree:$4$
Tame degree:$2$
Wild slopes:None
Galois mean slope:$1/2$
Galois splitting model:Not computed