Properties

Label 19.6.5.4
Base \(\Q_{19}\)
Degree \(6\)
e \(6\)
f \(1\)
c \(5\)
Galois group $C_6$

Related objects

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Defining polynomial

\( x^{6} + 76 \)

Invariants

Base field: $\Q_{19}$
Degree $d$ : $6$
Ramification exponent $e$ : $6$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $5$
Discriminant root field: $\Q_{19}(\sqrt{19*})$
Root number: $-i$
$|\Gal(K/\Q_{ 19 })|$: $6$
This field is Galois and abelian over $\Q_{19}$.

Intermediate fields

$\Q_{19}(\sqrt{19*})$, 19.3.2.1

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{19}$
Relative Eisenstein polynomial:\( x^{6} + 76 \)

Invariants of the Galois closure

Galois group:$C_6$
Inertia group:$C_6$
Unramified degree:$1$
Tame degree:$6$
Wild slopes:None
Galois Mean Slope:$5/6$
Global Splitting Model:\( x^{6} - x^{5} + 2 x^{4} - 11 x^{3} + 341 x^{2} - 708 x + 2876 \)