Properties

Label 13.12.8.3
Base \(\Q_{13}\)
Degree \(12\)
e \(3\)
f \(4\)
c \(8\)
Galois group $C_{12}$ (as 12T1)

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Defining polynomial

\( x^{12} + 26 x^{9} + 845 x^{6} + 6591 x^{3} + 114244 \)

Invariants

Base field: $\Q_{13}$
Degree $d$ : $12$
Ramification exponent $e$ : $3$
Residue field degree $f$ : $4$
Discriminant exponent $c$ : $8$
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, 13.4.0.1, 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:13.4.0.1 $\cong \Q_{13}(t)$ where $t$ is a root of \( x^{4} + x^{2} - x + 2 \)
Relative Eisenstein polynomial:$ x^{3} - 13 t^{2} \in\Q_{13}(t)[x]$

Invariants of the Galois closure

Galois group:$C_{12}$ (as 12T1)
Inertia group:Intransitive group isomorphic to $C_3$
Unramified degree:$4$
Tame degree:$3$
Wild slopes:None
Galois mean slope:$2/3$
Galois splitting model:$x^{12} + 39 x^{10} - 91 x^{9} + 1521 x^{8} + 10647 x^{7} + 67600 x^{6} + 276822 x^{5} + 1667523 x^{4} + 4644458 x^{3} + 12595401 x^{2} + 29389269 x + 68574961$