Properties

Label 17.14.13.2
Base \(\Q_{17}\)
Degree \(14\)
e \(14\)
f \(1\)
c \(13\)
Galois group $F_7 \times C_2$ (as 14T7)

Related objects

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Defining polynomial

\(x^{14} + 51\)  Toggle raw display

Invariants

Base field: $\Q_{17}$
Degree $d$: $14$
Ramification exponent $e$: $14$
Residue field degree $f$: $1$
Discriminant exponent $c$: $13$
Discriminant root field: $\Q_{17}(\sqrt{17\cdot 3})$
Root number: $1$
$|\Aut(K/\Q_{ 17 })|$: $2$
This field is not Galois over $\Q_{17}.$

Intermediate fields

$\Q_{17}(\sqrt{17\cdot 3})$, 17.7.6.1

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{17}$
Relative Eisenstein polynomial:\( x^{14} + 51 \)  Toggle raw display

Invariants of the Galois closure

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