Properties

Label 13.9.8.3
Base \(\Q_{13}\)
Degree \(9\)
e \(9\)
f \(1\)
c \(8\)
Galois group $C_9:C_3$ (as 9T6)

Related objects

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Defining polynomial

\( x^{9} - 52 \)

Invariants

Base field: $\Q_{13}$
Degree $d$ : $9$
Ramification exponent $e$ : $9$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $8$
Discriminant root field: $\Q_{13}$
Root number: $1$
$|\Aut(K/\Q_{ 13 })|$: $3$
This field is not Galois over $\Q_{13}$.

Intermediate fields

13.3.2.3

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{13}$
Relative Eisenstein polynomial:\( x^{9} - 52 \)

Invariants of the Galois closure

Galois group:$C_9:C_3$ (as 9T6)
Inertia group:$C_9$
Unramified degree:$3$
Tame degree:$9$
Wild slopes:None
Galois mean slope:$8/9$
Galois splitting model:$x^{9} - 117 x^{7} + 4563 x^{5} - 65910 x^{3} + 257049 x - 199927$