Properties

Label 7.12.6.1
Base \(\Q_{7}\)
Degree \(12\)
e \(2\)
f \(6\)
c \(6\)
Galois group $C_6\times C_2$ (as 12T2)

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Defining polynomial

\( x^{12} + 294 x^{8} + 3430 x^{6} + 21609 x^{4} + 487403 x^{2} + 2941225 \)

Invariants

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

Intermediate fields

$\Q_{7}(\sqrt{*})$, $\Q_{7}(\sqrt{7})$, $\Q_{7}(\sqrt{7*})$, 7.3.0.1, 7.4.2.1, 7.6.0.1, 7.6.3.1, 7.6.3.2

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

Unramified/totally ramified tower

Unramified subfield:7.6.0.1 $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{6} + 3 x^{2} - x + 5 \)
Relative Eisenstein polynomial:$ x^{2} - 7 t^{2} \in\Q_{7}(t)[x]$

Invariants of the Galois closure

Galois group:$C_2\times C_6$ (as 12T2)
Inertia group:Intransitive group isomorphic to $C_2$
Unramified degree:$6$
Tame degree:$2$
Wild slopes:None
Galois mean slope:$1/2$
Galois splitting model:$x^{12} - x^{11} + 17 x^{10} - 10 x^{9} + 130 x^{8} - 53 x^{7} + 485 x^{6} - 86 x^{5} + 884 x^{4} - 128 x^{3} + 576 x^{2} + 96 x + 64$