Properties

Label 13.6.0.1
Base \(\Q_{13}\)
Degree \(6\)
e \(1\)
f \(6\)
c \(0\)
Galois group $C_6$ (as 6T1)

Related objects

Learn more about

Defining polynomial

\( x^{6} + x^{2} - 2 x + 2 \)

Invariants

Base field: $\Q_{13}$
Degree $d$ : $6$
Ramification exponent $e$ : $1$
Residue field degree $f$ : $6$
Discriminant exponent $c$ : $0$
Discriminant root field: $\Q_{13}(\sqrt{*})$
Root number: $1$
$|\Gal(K/\Q_{ 13 })|$: $6$
This field is Galois and abelian over $\Q_{13}$.

Intermediate fields

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

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

Unramified/totally ramified tower

Unramified subfield:13.6.0.1 $\cong \Q_{13}(t)$ where $t$ is a root of \( x^{6} + x^{2} - 2 x + 2 \)
Relative Eisenstein polynomial:$ x - 13 \in\Q_{13}(t)[x]$

Invariants of the Galois closure

Galois group:$C_6$ (as 6T1)
Inertia group:Trivial
Unramified degree:$6$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x^{6} - 9 x^{4} - 4 x^{3} + 9 x^{2} + 3 x - 1$