Properties

Label 17.12.9.4
Base \(\Q_{17}\)
Degree \(12\)
e \(4\)
f \(3\)
c \(9\)
Galois group $C_{12}$ (as 12T1)

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Defining polynomial

\( x^{12} + 153 x^{8} + 7514 x^{4} + 132651 \)

Invariants

Base field: $\Q_{17}$
Degree $d$ : $12$
Ramification exponent $e$ : $4$
Residue field degree $f$ : $3$
Discriminant exponent $c$ : $9$
Discriminant root field: $\Q_{17}(\sqrt{17*})$
Root number: $-1$
$|\Gal(K/\Q_{ 17 })|$: $12$
This field is Galois and abelian over $\Q_{17}$.

Intermediate fields

$\Q_{17}(\sqrt{17*})$, 17.3.0.1, 17.4.3.3, 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:17.3.0.1 $\cong \Q_{17}(t)$ where $t$ is a root of \( x^{3} - x + 3 \)
Relative Eisenstein polynomial:$ x^{4} - 17 t^{3} \in\Q_{17}(t)[x]$

Invariants of the Galois closure

Galois group:$C_{12}$ (as 12T1)
Inertia group:Intransitive group isomorphic to $C_4$
Unramified degree:$3$
Tame degree:$4$
Wild slopes:None
Galois mean slope:$3/4$
Galois splitting model:Not computed