Properties

Label 7.12.8.2
Base \(\Q_{7}\)
Degree \(12\)
e \(3\)
f \(4\)
c \(8\)
Galois group $C_{12}$ (as 12T1)

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Defining polynomial

\( x^{12} + 49 x^{6} - 1029 x^{3} + 12005 \)

Invariants

Base field: $\Q_{7}$
Degree $d$ : $12$
Ramification exponent $e$ : $3$
Residue field degree $f$ : $4$
Discriminant exponent $c$ : $8$
Discriminant root field: $\Q_{7}(\sqrt{*})$
Root number: $1$
$|\Gal(K/\Q_{ 7 })|$: $12$
This field is Galois and abelian over $\Q_{7}$.

Intermediate fields

$\Q_{7}(\sqrt{*})$, 7.3.2.1, 7.4.0.1, 7.6.4.2

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

Unramified/totally ramified tower

Unramified subfield:7.4.0.1 $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{4} + x^{2} - 3 x + 5 \)
Relative Eisenstein polynomial:$ x^{3} - 7 t \in\Q_{7}(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:$x^{12} - x^{11} + x^{10} - 27 x^{9} + 27 x^{8} + 90 x^{7} + 53 x^{6} - 1353 x^{5} + 768 x^{4} + 3886 x^{3} + 1600 x^{2} - 5409 x + 1847$