Properties

Label 13.12.0.1
Base \(\Q_{13}\)
Degree \(12\)
e \(1\)
f \(12\)
c \(0\)
Galois group $C_{12}$ (as 12T1)

Related objects

Learn more about

Defining polynomial

\( x^{12} + x^{2} - x + 2 \)

Invariants

Base field: $\Q_{13}$
Degree $d$ : $12$
Ramification exponent $e$ : $1$
Residue field degree $f$ : $12$
Discriminant exponent $c$ : $0$
Discriminant root field: $\Q_{13}(\sqrt{*})$
Root number: $1$
$|\Gal(K/\Q_{ 13 })|$: $12$
This field is Galois and abelian over $\Q_{13}$.

Intermediate fields

$\Q_{13}(\sqrt{*})$, 13.3.0.1, 13.4.0.1, 13.6.0.1

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

Unramified/totally ramified tower

Unramified subfield:13.12.0.1 $\cong \Q_{13}(t)$ where $t$ is a root of \( x^{12} + x^{2} - x + 2 \)
Relative Eisenstein polynomial:$ x - 13 \in\Q_{13}(t)[x]$

Invariants of the Galois closure

Galois group:$C_{12}$ (as 12T1)
Inertia group:Trivial
Unramified degree:$12$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x^{12} + 3 x^{10} - x^{9} + 9 x^{8} + 9 x^{7} + 28 x^{6} + 18 x^{5} + 75 x^{4} + 26 x^{3} + 9 x^{2} + 3 x + 1$