Properties

Label 19.6.0.1
Base \(\Q_{19}\)
Degree \(6\)
e \(1\)
f \(6\)
c \(0\)
Galois group $C_6$ (as 6T1)

Related objects

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Defining polynomial

\( x^{6} - x + 3 \)

Invariants

Base field: $\Q_{19}$
Degree $d$ : $6$
Ramification exponent $e$ : $1$
Residue field degree $f$ : $6$
Discriminant exponent $c$ : $0$
Discriminant root field: $\Q_{19}(\sqrt{*})$
Root number: $1$
$|\Gal(K/\Q_{ 19 })|$: $6$
This field is Galois and abelian over $\Q_{19}$.

Intermediate fields

$\Q_{19}(\sqrt{*})$, 19.3.0.1

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

Unramified/totally ramified tower

Unramified subfield:19.6.0.1 $\cong \Q_{19}(t)$ where $t$ is a root of \( x^{6} - x + 3 \)
Relative Eisenstein polynomial:$ x - 19 \in\Q_{19}(t)[x]$

Invariants of the Galois closure

Galois group:$C_6$ (as 6T1)
Inertia group:Trivial
Unramified degree:$6$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$