Properties

Label 19.15.10.2
Base \(\Q_{19}\)
Degree \(15\)
e \(3\)
f \(5\)
c \(10\)
Galois group $C_{15}$ (as 15T1)

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Defining polynomial

\(x^{15} + 651605 x^{3} - 42093683\) Copy content Toggle raw display

Invariants

Base field: $\Q_{19}$
Degree $d$: $15$
Ramification exponent $e$: $3$
Residue field degree $f$: $5$
Discriminant exponent $c$: $10$
Discriminant root field: $\Q_{19}$
Root number: $1$
$\card{ \Gal(K/\Q_{ 19 }) }$: $15$
This field is Galois and abelian over $\Q_{19}.$
Visible slopes:None

Intermediate fields

19.3.2.1, 19.5.0.1

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

Unramified/totally ramified tower

Unramified subfield:19.5.0.1 $\cong \Q_{19}(t)$ where $t$ is a root of \( x^{5} + 5 x + 17 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{3} + 19 t \) $\ \in\Q_{19}(t)[x]$ 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:$C_{15}$ (as 15T1)
Inertia group:Intransitive group isomorphic to $C_3$
Wild inertia group:$C_1$
Unramified degree:$5$
Tame degree:$3$
Wild slopes:None
Galois mean slope:$2/3$
Galois splitting model:Not computed