Properties

Label 19.10.8.1
Base \(\Q_{19}\)
Degree \(10\)
e \(5\)
f \(2\)
c \(8\)
Galois group $D_5$ (as 10T2)

Related objects

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Defining polynomial

\( x^{10} - 209 x^{5} + 11552 \)

Invariants

Base field: $\Q_{19}$
Degree $d$ : $10$
Ramification exponent $e$ : $5$
Residue field degree $f$ : $2$
Discriminant exponent $c$ : $8$
Discriminant root field: $\Q_{19}(\sqrt{*})$
Root number: $1$
$|\Gal(K/\Q_{ 19 })|$: $10$
This field is Galois over $\Q_{19}$.

Intermediate fields

$\Q_{19}(\sqrt{*})$, 19.5.4.1 x5

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{19}(\sqrt{*})$ $\cong \Q_{19}(t)$ where $t$ is a root of \( x^{2} - x + 2 \)
Relative Eisenstein polynomial:$ x^{5} - 19 t^{5} \in\Q_{19}(t)[x]$

Invariants of the Galois closure

Galois group:$D_5$ (as 10T2)
Inertia group:Intransitive group isomorphic to $C_5$
Unramified degree:$2$
Tame degree:$5$
Wild slopes:None
Galois mean slope:$4/5$
Galois splitting model:Not computed