Properties

Label 19.9.6.3
Base \(\Q_{19}\)
Degree \(9\)
e \(3\)
f \(3\)
c \(6\)
Galois group $C_9$ (as 9T1)

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Defining polynomial

\(x^{9} - 152 x^{6} + 5776 x^{3} + 1982251\) Copy content Toggle raw display

Invariants

Base field: $\Q_{19}$
Degree $d$: $9$
Ramification exponent $e$: $3$
Residue field degree $f$: $3$
Discriminant exponent $c$: $6$
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

19.3.0.1

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

Unramified/totally ramified tower

Unramified subfield:19.3.0.1 $\cong \Q_{19}(t)$ where $t$ is a root of \( x^{3} + 4 x + 17 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{3} + 19 t^{2} \) $\ \in\Q_{19}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{2} + 3z + 3$
Associated inertia:$1$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$C_9$ (as 9T1)
Inertia group:Intransitive group isomorphic to $C_3$
Wild inertia group:$C_1$
Unramified degree:$3$
Tame degree:$3$
Wild slopes:None
Galois mean slope:$2/3$
Galois splitting model:Not computed