Properties

Label 3.14.12.1
Base \(\Q_{3}\)
Degree \(14\)
e \(7\)
f \(2\)
c \(12\)
Galois group $F_7$ (as 14T4)

Related objects

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Defining polynomial

\( x^{14} - 3 x^{7} + 18 \)

Invariants

Base field: $\Q_{3}$
Degree $d$: $14$
Ramification exponent $e$: $7$
Residue field degree $f$: $2$
Discriminant exponent $c$: $12$
Discriminant root field: $\Q_{3}(\sqrt{2})$
Root number: $1$
$|\Aut(K/\Q_{ 3 })|$: $2$
This field is not Galois over $\Q_{3}.$

Intermediate fields

$\Q_{3}(\sqrt{2})$, 3.7.6.1

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

Unramified/totally ramified tower

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

Invariants of the Galois closure

Galois group:$F_7$ (as 14T4)
Inertia group:Intransitive group isomorphic to $C_7$
Unramified degree:$6$
Tame degree:$7$
Wild slopes:None
Galois mean slope:$6/7$
Galois splitting model:Not computed