Properties

Label 3.14.13.2
Base \(\Q_{3}\)
Degree \(14\)
e \(14\)
f \(1\)
c \(13\)
Galois group $F_7 \times C_2$ (as 14T7)

Related objects

Downloads

Learn more

Defining polynomial

\(x^{14} + 3\) Copy content Toggle raw display

Invariants

Base field: $\Q_{3}$
Degree $d$: $14$
Ramification exponent $e$: $14$
Residue field degree $f$: $1$
Discriminant exponent $c$: $13$
Discriminant root field: $\Q_{3}(\sqrt{3\cdot 2})$
Root number: $i$
$\card{ \Aut(K/\Q_{ 3 }) }$: $2$
This field is not Galois over $\Q_{3}.$
Visible slopes:None

Intermediate fields

$\Q_{3}(\sqrt{3\cdot 2})$, 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:$\Q_{3}$
Relative Eisenstein polynomial: \( x^{14} + 3 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{13} + 2z^{12} + z^{11} + z^{10} + 2z^{9} + z^{8} + z^{4} + 2z^{3} + z^{2} + z + 2$
Associated inertia:$6$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$C_2\times F_7$ (as 14T7)
Inertia group:$C_{14}$ (as 14T1)
Wild inertia group:$C_1$
Unramified degree:$6$
Tame degree:$14$
Wild slopes:None
Galois mean slope:$13/14$
Galois splitting model:Not computed