Properties

Label 7.12.10.4
Base \(\Q_{7}\)
Degree \(12\)
e \(6\)
f \(2\)
c \(10\)
Galois group $C_{12}$ (as 12T1)

Related objects

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Defining polynomial

\( x^{12} - 7 x^{6} + 147 \)

Invariants

Base field: $\Q_{7}$
Degree $d$ : $12$
Ramification exponent $e$ : $6$
Residue field degree $f$ : $2$
Discriminant exponent $c$ : $10$
Discriminant root field: $\Q_{7}(\sqrt{*})$
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, 7.4.2.2, 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:$\Q_{7}(\sqrt{*})$ $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{2} - x + 3 \)
Relative Eisenstein polynomial:$ x^{6} - 7 t \in\Q_{7}(t)[x]$

Invariants of the Galois closure

Galois group:$C_{12}$ (as 12T1)
Inertia group:Intransitive group isomorphic to $C_6$
Unramified degree:$2$
Tame degree:$6$
Wild slopes:None
Galois mean slope:$5/6$
Galois splitting model:$x^{12} - x^{11} - 38 x^{10} - 14 x^{9} + 495 x^{8} + 688 x^{7} - 2157 x^{6} - 5123 x^{5} - 25 x^{4} + 7175 x^{3} + 4629 x^{2} - 534 x - 727$