Properties

Label 3.14.7.1
Base \(\Q_{3}\)
Degree \(14\)
e \(2\)
f \(7\)
c \(7\)
Galois group $C_{14}$ (as 14T1)

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Defining polynomial

\( x^{14} - 54 x^{8} - 243 x^{4} - 729 x^{2} - 2187 \)

Invariants

Base field: $\Q_{3}$
Degree $d$: $14$
Ramification exponent $e$: $2$
Residue field degree $f$: $7$
Discriminant exponent $c$: $7$
Discriminant root field: $\Q_{3}(\sqrt{3})$
Root number: $i$
$|\Gal(K/\Q_{ 3 })|$: $14$
This field is Galois and abelian over $\Q_{3}.$

Intermediate fields

$\Q_{3}(\sqrt{3})$, 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:3.7.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{7} + x^{2} - x + 1 \)
Relative Eisenstein polynomial:$ x^{2} - 3 t^{2} \in\Q_{3}(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