Properties

Label 7.14.0.1
Base \(\Q_{7}\)
Degree \(14\)
e \(1\)
f \(14\)
c \(0\)
Galois group $C_{14}$ (as 14T1)

Related objects

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Defining polynomial

\(x^{14} + 2 x^{2} - 2 x + 3\)  Toggle raw display

Invariants

Base field: $\Q_{7}$
Degree $d$: $14$
Ramification exponent $e$: $1$
Residue field degree $f$: $14$
Discriminant exponent $c$: $0$
Discriminant root field: $\Q_{7}(\sqrt{3})$
Root number: $1$
$|\Gal(K/\Q_{ 7 })|$: $14$
This field is Galois and abelian over $\Q_{7}.$

Intermediate fields

$\Q_{7}(\sqrt{3})$, 7.7.0.1

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

Unramified/totally ramified tower

Unramified subfield:7.14.0.1 $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{14} + 2 x^{2} - 2 x + 3 \)  Toggle raw display
Relative Eisenstein polynomial:\( x - 7 \)$\ \in\Q_{7}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_{14}$ (as 14T1)
Inertia group:trivial
Wild inertia group:$C_1$
Unramified degree:$14$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x^{14} + 61 x^{12} + 1402 x^{10} + 15247 x^{8} + 80418 x^{6} + 185798 x^{4} + 127429 x^{2} + 18769$