Properties

Label 19.6.5.1
Base \(\Q_{19}\)
Degree \(6\)
e \(6\)
f \(1\)
c \(5\)
Galois group $C_6$ (as 6T1)

Related objects

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Defining polynomial

\( x^{6} - 304 \)

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

$\Q_{19}(\sqrt{19})$, 19.3.2.3

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:\( x^{6} - 304 \)

Invariants of the Galois closure

Galois group:$C_6$ (as 6T1)
Inertia group:$C_6$
Unramified degree:$1$
Tame degree:$6$
Wild slopes:None
Galois mean slope:$5/6$
Galois splitting model:$x^{6} - x^{5} - 55 x^{4} + 160 x^{3} + 246 x^{2} - 1107 x + 729$