Properties

Label 17.14.7.1
Base \(\Q_{17}\)
Degree \(14\)
e \(2\)
f \(7\)
c \(7\)
Galois group $C_{14}$ (as 14T1)

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Defining polynomial

\( x^{14} - 9826 x^{8} + 24137569 x^{2} - 3693048057 \)

Invariants

Base field: $\Q_{17}$
Degree $d$: $14$
Ramification exponent $e$: $2$
Residue field degree $f$: $7$
Discriminant exponent $c$: $7$
Discriminant root field: $\Q_{17}(\sqrt{17})$
Root number: $1$
$|\Gal(K/\Q_{ 17 })|$: $14$
This field is Galois and abelian over $\Q_{17}.$

Intermediate fields

$\Q_{17}(\sqrt{17})$, 17.7.0.1

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

Unramified/totally ramified tower

Unramified subfield:17.7.0.1 $\cong \Q_{17}(t)$ where $t$ is a root of \( x^{7} - x + 3 \)
Relative Eisenstein polynomial:$ x^{2} - 17 t^{2} \in\Q_{17}(t)[x]$

Invariants of the Galois closure

Galois group:$C_{14}$ (as 14T1)
Inertia group:Intransitive group isomorphic to $C_2$
Unramified degree:$7$
Tame degree:$2$
Wild slopes:None
Galois mean slope:$1/2$
Galois splitting model:Not computed