Properties

Label 13.6.3.2
Base \(\Q_{13}\)
Degree \(6\)
e \(2\)
f \(3\)
c \(3\)
Galois group $C_6$ (as 6T1)

Related objects

Learn more about

Defining polynomial

\( x^{6} - 338 x^{2} + 13182 \)

Invariants

Base field: $\Q_{13}$
Degree $d$ : $6$
Ramification exponent $e$ : $2$
Residue field degree $f$ : $3$
Discriminant exponent $c$ : $3$
Discriminant root field: $\Q_{13}(\sqrt{13*})$
Root number: $-1$
$|\Gal(K/\Q_{ 13 })|$: $6$
This field is Galois and abelian over $\Q_{13}$.

Intermediate fields

$\Q_{13}(\sqrt{13*})$, 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:13.3.0.1 $\cong \Q_{13}(t)$ where $t$ is a root of \( x^{3} - 2 x + 6 \)
Relative Eisenstein polynomial:$ x^{2} - 13 t \in\Q_{13}(t)[x]$

Invariants of the Galois closure

Galois group:$C_6$ (as 6T1)
Inertia group:Intransitive group isomorphic to $C_2$
Unramified degree:$3$
Tame degree:$2$
Wild slopes:None
Galois mean slope:$1/2$
Galois splitting model:$x^{6} - 3 x^{5} - 51 x^{4} + 105 x^{3} + 723 x^{2} - 867 x - 2609$