Properties

Label 19.12.10.2
Base \(\Q_{19}\)
Degree \(12\)
e \(6\)
f \(2\)
c \(10\)
Galois group $C_6\times C_2$ (as 12T2)

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Defining polynomial

\( x^{12} + 57 x^{6} + 1444 \)

Invariants

Base field: $\Q_{19}$
Degree $d$ : $12$
Ramification exponent $e$ : $6$
Residue field degree $f$ : $2$
Discriminant exponent $c$ : $10$
Discriminant root field: $\Q_{19}$
Root number: $1$
$|\Gal(K/\Q_{ 19 })|$: $12$
This field is Galois and abelian over $\Q_{19}$.

Intermediate fields

$\Q_{19}(\sqrt{*})$, $\Q_{19}(\sqrt{19})$, $\Q_{19}(\sqrt{19*})$, 19.3.2.3, 19.4.2.1, 19.6.4.1, 19.6.5.1, 19.6.5.6

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^{6} - 19 t^{2} \in\Q_{19}(t)[x]$

Invariants of the Galois closure

Galois group:$C_2\times C_6$ (as 12T2)
Inertia group:Intransitive group isomorphic to $C_6$
Unramified degree:$2$
Tame degree:$6$
Wild slopes:None
Galois mean slope:$5/6$
Galois splitting model:Not computed