Properties

Label 7.12.10.5
Base \(\Q_{7}\)
Degree \(12\)
e \(6\)
f \(2\)
c \(10\)
Galois group $C_{12}$ (as 12T1)

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Defining polynomial

\(x^{12} + 56 x^{6} + 1323\)  Toggle raw display

Invariants

Base field: $\Q_{7}$
Degree $d$: $12$
Ramification exponent $e$: $6$
Residue field degree $f$: $2$
Discriminant exponent $c$: $10$
Discriminant root field: $\Q_{7}(\sqrt{3})$
Root number: $-1$
$|\Gal(K/\Q_{ 7 })|$: $12$
This field is Galois and abelian over $\Q_{7}.$

Intermediate fields

$\Q_{7}(\sqrt{3})$, 7.3.2.2, 7.4.2.2, 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:$\Q_{7}(\sqrt{3})$ $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{2} - x + 3 \)  Toggle raw display
Relative Eisenstein polynomial:\( x^{6} - 7 t^{3} \)$\ \in\Q_{7}(t)[x]$  Toggle raw display

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:$x^{12} - x^{11} - 12 x^{10} + 11 x^{9} + 54 x^{8} - 43 x^{7} - 113 x^{6} + 71 x^{5} + 110 x^{4} - 46 x^{3} - 40 x^{2} + 8 x + 1$