Properties

Label 19.6.4.1
Base \(\Q_{19}\)
Degree \(6\)
e \(3\)
f \(2\)
c \(4\)
Galois group $C_6$ (as 6T1)

Related objects

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Defining polynomial

\( x^{6} + 57 x^{3} + 1444 \)

Invariants

Base field: $\Q_{19}$
Degree $d$ : $6$
Ramification exponent $e$ : $3$
Residue field degree $f$ : $2$
Discriminant exponent $c$ : $4$
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

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

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{19}(\sqrt{*})$ $\cong \Q_{19}(t)$ where $t$ is a root of \( x^{2} - x + 2 \)
Relative Eisenstein polynomial:$ x^{3} - 19 t^{2} \in\Q_{19}(t)[x]$

Invariants of the Galois closure

Galois group:$C_6$ (as 6T1)
Inertia group:Intransitive group isomorphic to $C_3$
Unramified degree:$2$
Tame degree:$3$
Wild slopes:None
Galois mean slope:$2/3$
Galois splitting model:$x^{6} - x^{5} + x^{4} - 99 x^{3} + 281 x^{2} + 573 x + 981$