Properties

Label 19.13.12.1
Base \(\Q_{19}\)
Degree \(13\)
e \(13\)
f \(1\)
c \(12\)
Galois group $F_{13}$ (as 13T6)

Related objects

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Defining polynomial

\( x^{13} - 19 \)

Invariants

Base field: $\Q_{19}$
Degree $d$ : $13$
Ramification exponent $e$ : $13$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $12$
Discriminant root field: $\Q_{19}(\sqrt{*})$
Root number: $1$
$|\Aut(K/\Q_{ 19 })|$: $1$
This field is not Galois over $\Q_{19}$.

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 19 }$.

Unramified/totally ramified tower

Unramified subfield:$\Q_{19}$
Relative Eisenstein polynomial:\( x^{13} - 19 \)

Invariants of the Galois closure

Galois group:$F_{13}$ (as 13T6)
Inertia group:$C_{13}$
Unramified degree:$12$
Tame degree:$13$
Wild slopes:None
Galois mean slope:$12/13$
Galois splitting model:$x^{13} - 52 x^{12} + 1248 x^{11} - 18304 x^{10} + 183040 x^{9} - 1317888 x^{8} + 7028736 x^{7} - 28114944 x^{6} + 84344832 x^{5} - 187432960 x^{4} + 299892736 x^{3} - 327155712 x^{2} + 218103808 x - 67108883$