Properties

Label 17.14.0.1
Base \(\Q_{17}\)
Degree \(14\)
e \(1\)
f \(14\)
c \(0\)
Galois group $C_{14}$ (as 14T1)

Related objects

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Defining polynomial

\( x^{14} + x^{2} - 2 x + 6 \)

Invariants

Base field: $\Q_{17}$
Degree $d$: $14$
Ramification exponent $e$: $1$
Residue field degree $f$: $14$
Discriminant exponent $c$: $0$
Discriminant root field: $\Q_{17}(\sqrt{3})$
Root number: $1$
$|\Gal(K/\Q_{ 17 })|$: $14$
This field is Galois and abelian over $\Q_{17}.$

Intermediate fields

$\Q_{17}(\sqrt{3})$, 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:17.14.0.1 $\cong \Q_{17}(t)$ where $t$ is a root of \( x^{14} + x^{2} - 2 x + 6 \)
Relative Eisenstein polynomial:$ x - 17 \in\Q_{17}(t)[x]$

Invariants of the Galois closure

Galois group:$C_{14}$ (as 14T1)
Inertia group:trivial
Unramified degree:$14$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x^{14} - x^{13} - 52 x^{12} + 31 x^{11} + 908 x^{10} - 162 x^{9} - 6351 x^{8} - 856 x^{7} + 16495 x^{6} + 3600 x^{5} - 18257 x^{4} - 3381 x^{3} + 8945 x^{2} + 874 x - 1583$