Base \(\Q_{19}\)
Degree \(15\)
e \(5\)
f \(3\)
c \(12\)
Galois group $D_5\times C_3$ (as 15T3)

Related objects

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Defining polynomial

\(x^{15} - 361 x^{5} + 27436\)  Toggle raw display


Base field: $\Q_{19}$
Degree $d$: $15$
Ramification exponent $e$: $5$
Residue field degree $f$: $3$
Discriminant exponent $c$: $12$
Discriminant root field: $\Q_{19}$
Root number: $1$
$|\Aut(K/\Q_{ 19 })|$: $3$
This field is not Galois over $\Q_{19}.$

Intermediate fields,

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

Unramified/totally ramified tower

Unramified subfield: $\cong \Q_{19}(t)$ where $t$ is a root of \( x^{3} - x + 4 \)  Toggle raw display
Relative Eisenstein polynomial:\( x^{5} - 19 t \)$\ \in\Q_{19}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_3\times D_5$ (as 15T3)
Inertia group:Intransitive group isomorphic to $C_5$
Wild inertia group:$C_1$
Unramified degree:$6$
Tame degree:$5$
Wild slopes:None
Galois mean slope:$4/5$
Galois splitting model:Not computed