Properties

Label 19.12.11.3
Base \(\Q_{19}\)
Degree \(12\)
e \(12\)
f \(1\)
c \(11\)
Galois group $D_4 \times C_3$ (as 12T14)

Related objects

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Defining polynomial

\( x^{12} + 19456 \)

Invariants

Base field: $\Q_{19}$
Degree $d$ : $12$
Ramification exponent $e$ : $12$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $11$
Discriminant root field: $\Q_{19}(\sqrt{19})$
Root number: $i$
$|\Aut(K/\Q_{ 19 })|$: $6$
This field is not Galois over $\Q_{19}$.

Intermediate fields

$\Q_{19}(\sqrt{19*})$, 19.3.2.3, 19.4.3.1, 19.6.5.6

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:\( x^{12} + 19456 \)

Invariants of the Galois closure

Galois group:$C_3\times D_4$ (as 12T14)
Inertia group:$C_{12}$
Unramified degree:$2$
Tame degree:$12$
Wild slopes:None
Galois mean slope:$11/12$
Galois splitting model:Not computed