Properties

Label 19.9.8.6
Base \(\Q_{19}\)
Degree \(9\)
e \(9\)
f \(1\)
c \(8\)
Galois group $C_9$ (as 9T1)

Related objects

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Defining polynomial

\(x^{9} + 1216\)  Toggle raw display

Invariants

Base field: $\Q_{19}$
Degree $d$: $9$
Ramification exponent $e$: $9$
Residue field degree $f$: $1$
Discriminant exponent $c$: $8$
Discriminant root field: $\Q_{19}$
Root number: $1$
$|\Gal(K/\Q_{ 19 })|$: $9$
This field is Galois and abelian over $\Q_{19}.$

Intermediate fields

19.3.2.2

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{19}$
Relative Eisenstein polynomial:\( x^{9} + 1216 \)  Toggle raw display

Invariants of the Galois closure

Galois group:$C_9$ (as 9T1)
Inertia group:$C_9$
Unramified degree:$1$
Tame degree:$9$
Wild slopes:None
Galois mean slope:$8/9$
Galois splitting model:Not computed