Properties

Label 19.6.3.1
Base \(\Q_{19}\)
Degree \(6\)
e \(2\)
f \(3\)
c \(3\)
Galois group $C_6$ (as 6T1)

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Defining polynomial

\( x^{6} - 38 x^{4} + 361 x^{2} - 109744 \)

Invariants

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

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

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

Unramified/totally ramified tower

Unramified subfield:19.3.0.1 $\cong \Q_{19}(t)$ where $t$ is a root of \( x^{3} - x + 4 \)
Relative Eisenstein polynomial:$ x^{2} - 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 $C_2$
Unramified degree:$3$
Tame degree:$2$
Wild slopes:None
Galois mean slope:$1/2$
Galois splitting model:$x^{6} - x^{5} - 34 x^{4} + 34 x^{3} + 316 x^{2} - 316 x - 559$