Properties

Label 7.12.8.1
Base \(\Q_{7}\)
Degree \(12\)
e \(3\)
f \(4\)
c \(8\)
Galois group $C_{12}$ (as 12T1)

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Defining polynomial

\(x^{12} + 15 x^{10} + 40 x^{9} + 84 x^{8} + 120 x^{7} + 53 x^{6} + 414 x^{5} - 1293 x^{4} - 1830 x^{3} + 10968 x^{2} - 13836 x + 12004\) Copy content Toggle raw display

Invariants

Base field: $\Q_{7}$
Degree $d$: $12$
Ramification exponent $e$: $3$
Residue field degree $f$: $4$
Discriminant exponent $c$: $8$
Discriminant root field: $\Q_{7}(\sqrt{3})$
Root number: $1$
$\card{ \Gal(K/\Q_{ 7 }) }$: $12$
This field is Galois and abelian over $\Q_{7}.$
Visible slopes:None

Intermediate fields

$\Q_{7}(\sqrt{3})$, 7.3.2.2, 7.4.0.1, 7.6.4.3

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 \( x^{4} + 5 x^{2} + 4 x + 3 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{3} + 7 \) $\ \in\Q_{7}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Data not computed

Invariants of the Galois closure

Galois group: $C_{12}$ (as 12T1)
Inertia group: Intransitive group isomorphic to $C_3$
Wild inertia group: $C_1$
Unramified degree: $4$
Tame degree: $3$
Wild slopes: None
Galois mean slope: $2/3$
Galois splitting model:$x^{12} - x^{11} + 3 x^{10} - 4 x^{9} + 9 x^{8} + 2 x^{7} + 12 x^{6} + x^{5} + 25 x^{4} - 11 x^{3} + 5 x^{2} - 2 x + 1$