Properties

Label 7.12.10.3
Base \(\Q_{7}\)
Degree \(12\)
e \(6\)
f \(2\)
c \(10\)
Galois group $C_6\times C_2$ (as 12T2)

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Defining polynomial

\( x^{12} - 49 x^{6} + 3969 \)

Invariants

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

Intermediate fields

$\Q_{7}(\sqrt{*})$, $\Q_{7}(\sqrt{7})$, $\Q_{7}(\sqrt{7*})$, 7.3.2.1, 7.4.2.1, 7.6.4.2, 7.6.5.3, 7.6.5.4

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^{4} \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 $C_6$
Unramified degree:$2$
Tame degree:$6$
Wild slopes:None
Galois mean slope:$5/6$
Galois splitting model:$x^{12} - x^{11} + 4 x^{10} - 49 x^{9} + 117 x^{8} - 40 x^{7} - 113 x^{6} - 1035 x^{5} + 1872 x^{4} + 1393 x^{3} + 1759 x^{2} - 2452 x + 4432$