Properties

Label 3.14.0.1
Base \(\Q_{3}\)
Degree \(14\)
e \(1\)
f \(14\)
c \(0\)
Galois group $C_{14}$ (as 14T1)

Related objects

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Defining polynomial

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

Invariants

Base field: $\Q_{3}$
Degree $d$: $14$
Ramification exponent $e$: $1$
Residue field degree $f$: $14$
Discriminant exponent $c$: $0$
Discriminant root field: $\Q_{3}(\sqrt{2})$
Root number: $1$
$|\Gal(K/\Q_{ 3 })|$: $14$
This field is Galois and abelian over $\Q_{3}.$

Intermediate fields

$\Q_{3}(\sqrt{2})$, 3.7.0.1

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

Unramified/totally ramified tower

Unramified subfield:3.14.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{14} - x + 2 \)
Relative Eisenstein polynomial:$ x - 3 \in\Q_{3}(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:$x^{14} - x^{13} - 13 x^{12} + 12 x^{11} + 66 x^{10} - 55 x^{9} - 165 x^{8} + 120 x^{7} + 210 x^{6} - 126 x^{5} - 126 x^{4} + 56 x^{3} + 28 x^{2} - 7 x - 1$