Properties

Label 17.12.6.1
Base \(\Q_{17}\)
Degree \(12\)
e \(2\)
f \(6\)
c \(6\)
Galois group $C_6\times C_2$ (as 12T2)

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Defining polynomial

\( x^{12} + 117912 x^{6} - 1419857 x^{2} + 3475809936 \)

Invariants

Base field: $\Q_{17}$
Degree $d$ : $12$
Ramification exponent $e$ : $2$
Residue field degree $f$ : $6$
Discriminant exponent $c$ : $6$
Discriminant root field: $\Q_{17}$
Root number: $-1$
$|\Gal(K/\Q_{ 17 })|$: $12$
This field is Galois and abelian over $\Q_{17}$.

Intermediate fields

$\Q_{17}(\sqrt{*})$, $\Q_{17}(\sqrt{17})$, $\Q_{17}(\sqrt{17*})$, 17.3.0.1, 17.4.2.1, 17.6.0.1, 17.6.3.1, 17.6.3.2

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

Unramified/totally ramified tower

Unramified subfield:17.6.0.1 $\cong \Q_{17}(t)$ where $t$ is a root of \( x^{6} - x + 12 \)
Relative Eisenstein polynomial:$ x^{2} - 17 t^{2} \in\Q_{17}(t)[x]$

Invariants of the Galois closure

Galois group:$C_2\times C_6$ (as 12T2)
Inertia group:Intransitive group isomorphic to $C_2$
Unramified degree:$6$
Tame degree:$2$
Wild slopes:None
Galois mean slope:$1/2$
Galois splitting model:Not computed