Properties

Label 13.12.10.4
Base \(\Q_{13}\)
Degree \(12\)
e \(6\)
f \(2\)
c \(10\)
Galois group $C_{12}$ (as 12T1)

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Defining polynomial

\( x^{12} - 13 x^{6} + 338 \)

Invariants

Base field: $\Q_{13}$
Degree $d$ : $12$
Ramification exponent $e$ : $6$
Residue field degree $f$ : $2$
Discriminant exponent $c$ : $10$
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

$\Q_{13}(\sqrt{*})$, 13.3.2.3, 13.4.2.2, 13.6.4.2

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

Unramified/totally ramified tower

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

Invariants of the Galois closure

Galois group:$C_{12}$ (as 12T1)
Inertia group:Intransitive group isomorphic to $C_6$
Unramified degree:$2$
Tame degree:$6$
Wild slopes:None
Galois mean slope:$5/6$
Galois splitting model:$x^{12} + 39 x^{10} - 91 x^{9} - 234 x^{8} - 9243 x^{7} - 6695 x^{6} + 3042 x^{5} + 922233 x^{4} + 4370678 x^{3} + 28725606 x^{2} + 80259114 x + 161143021$