Properties

Label 17.12.11.2
Base \(\Q_{17}\)
Degree \(12\)
e \(12\)
f \(1\)
c \(11\)
Galois group $S_3 \times C_4$ (as 12T11)

Related objects

Learn more about

Defining polynomial

\( x^{12} - 153 \)

Invariants

Base field: $\Q_{17}$
Degree $d$ : $12$
Ramification exponent $e$ : $12$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $11$
Discriminant root field: $\Q_{17}(\sqrt{17})$
Root number: $-1$
$|\Aut(K/\Q_{ 17 })|$: $4$
This field is not Galois over $\Q_{17}$.

Intermediate fields

$\Q_{17}(\sqrt{17})$, 17.3.2.1, 17.4.3.2, 17.6.5.1

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{17}$
Relative Eisenstein polynomial:\( x^{12} - 153 \)

Invariants of the Galois closure

Galois group:$C_4\times S_3$ (as 12T11)
Inertia group:$C_{12}$
Unramified degree:$2$
Tame degree:$12$
Wild slopes:None
Galois mean slope:$11/12$
Galois splitting model:Not computed