Properties

Label 13.9.0.1
Base \(\Q_{13}\)
Degree \(9\)
e \(1\)
f \(9\)
c \(0\)
Galois group $C_9$ (as 9T1)

Related objects

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Defining polynomial

\(x^{9} - 2 x + 2\)  Toggle raw display

Invariants

Base field: $\Q_{13}$
Degree $d$: $9$
Ramification exponent $e$: $1$
Residue field degree $f$: $9$
Discriminant exponent $c$: $0$
Discriminant root field: $\Q_{13}$
Root number: $1$
$|\Gal(K/\Q_{ 13 })|$: $9$
This field is Galois and abelian over $\Q_{13}.$

Intermediate fields

13.3.0.1

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

Unramified/totally ramified tower

Unramified subfield:13.9.0.1 $\cong \Q_{13}(t)$ where $t$ is a root of \( x^{9} - 2 x + 2 \)  Toggle raw display
Relative Eisenstein polynomial:\( x - 13 \)$\ \in\Q_{13}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_9$ (as 9T1)
Inertia group:trivial
Wild inertia group:$C_1$
Unramified degree:$9$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x^{9} - x^{8} - 8 x^{7} + 7 x^{6} + 21 x^{5} - 15 x^{4} - 20 x^{3} + 10 x^{2} + 5 x - 1$