Properties

Label 19.9.8.4
Base \(\Q_{19}\)
Degree \(9\)
e \(9\)
f \(1\)
c \(8\)
Galois group $C_9$ (as 9T1)

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Defining polynomial

\(x^{9} + 114\) Copy content 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: $1$
$\card{ \Gal(K/\Q_{ 19 }) }$: $9$
This field is Galois and abelian over $\Q_{19}.$
Visible slopes:None

Intermediate fields

19.3.2.1

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{19}$
Relative Eisenstein polynomial: \( x^{9} + 114 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{8} + 9z^{7} + 17z^{6} + 8z^{5} + 12z^{4} + 12z^{3} + 8z^{2} + 17z + 9$
Associated inertia:$1$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$C_9$ (as 9T1)
Inertia group:$C_9$ (as 9T1)
Wild inertia group:$C_1$
Unramified degree:$1$
Tame degree:$9$
Wild slopes:None
Galois mean slope:$8/9$
Galois splitting model:Not computed