Properties

Label 19.12.11.1
Base \(\Q_{19}\)
Degree \(12\)
e \(12\)
f \(1\)
c \(11\)
Galois group $D_4 \times C_3$ (as 12T14)

Related objects

Learn more about

Defining polynomial

\(x^{12} + 76\)  Toggle raw display

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

$\Q_{19}(\sqrt{19\cdot 2})$, 19.3.2.1, 19.4.3.1, 19.6.5.4

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:\( x^{12} + 76 \)  Toggle raw display

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