Properties

Label 13.12.9.1
Base \(\Q_{13}\)
Degree \(12\)
e \(4\)
f \(3\)
c \(9\)
Galois group $C_{12}$ (as 12T1)

Related objects

Learn more about

Defining polynomial

\( x^{12} - 104 x^{8} - 45968 x^{4} - 2847312 \)

Invariants

Base field: $\Q_{13}$
Degree $d$ : $12$
Ramification exponent $e$ : $4$
Residue field degree $f$ : $3$
Discriminant exponent $c$ : $9$
Discriminant root field: $\Q_{13}(\sqrt{13})$
Root number: $-1$
$|\Gal(K/\Q_{ 13 })|$: $12$
This field is Galois and abelian over $\Q_{13}$.

Intermediate fields

$\Q_{13}(\sqrt{13})$, 13.3.0.1, 13.4.3.1, 13.6.3.1

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^{4} - 13 t^{4} \in\Q_{13}(t)[x]$

Invariants of the Galois closure

Galois group:$C_{12}$ (as 12T1)
Inertia group:Intransitive group isomorphic to $C_4$
Unramified degree:$3$
Tame degree:$4$
Wild slopes:None
Galois mean slope:$3/4$
Galois splitting model:$x^{12} - 3 x^{11} + 36 x^{10} - 6 x^{9} + 453 x^{8} + 1146 x^{7} + 3330 x^{6} + 13752 x^{5} + 37713 x^{4} - 14943 x^{3} + 108324 x^{2} - 10971 x + 141319$