Properties

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

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Defining polynomial

\(x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1484 x^{10} + 3752 x^{9} + 7448 x^{8} + 11782 x^{7} + 14938 x^{6} + 15008 x^{5} + 11452 x^{4} + 6328 x^{3} + 2632 x^{2} + 896 x + 185\) Copy content Toggle raw display

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: $1$
$\card{ \Aut(K/\Q_{ 3 }) }$: $2$
This field is not Galois over $\Q_{3}.$
Visible slopes:None

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 \( x^{2} + 2 x + 2 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{7} + 3 \) $\ \in\Q_{3}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{6} + z^{5} + 2z^{3} + 2z^{2} + 1$
Associated inertia:$3$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$F_7$ (as 14T4)
Inertia group:Intransitive group isomorphic to $C_7$
Wild inertia group:$C_1$
Unramified degree:$6$
Tame degree:$7$
Wild slopes:None
Galois mean slope:$6/7$
Galois splitting model:Not computed