Properties

Label 7.12.9.1
Base \(\Q_{7}\)
Degree \(12\)
e \(4\)
f \(3\)
c \(9\)
Galois group $D_4 \times C_3$ (as 12T14)

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Defining polynomial

\( x^{12} - 49 x^{4} + 686 \)

Invariants

Base field: $\Q_{7}$
Degree $d$ : $12$
Ramification exponent $e$ : $4$
Residue field degree $f$ : $3$
Discriminant exponent $c$ : $9$
Discriminant root field: $\Q_{7}(\sqrt{7})$
Root number: $-i$
$|\Aut(K/\Q_{ 7 })|$: $6$
This field is not Galois over $\Q_{7}$.

Intermediate fields

$\Q_{7}(\sqrt{7*})$, 7.3.0.1, 7.4.3.1, 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:7.3.0.1 $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{3} - x + 2 \)
Relative Eisenstein polynomial:$ x^{4} - 7 t \in\Q_{7}(t)[x]$

Invariants of the Galois closure

Galois group:$C_3\times D_4$ (as 12T14)
Inertia group:Intransitive group isomorphic to $C_4$
Unramified degree:$6$
Tame degree:$4$
Wild slopes:None
Galois mean slope:$3/4$
Galois splitting model:$x^{12} + 15 x^{10} - 4 x^{9} + 45 x^{8} - 3 x^{7} - 81 x^{6} + 36 x^{5} + 264 x^{4} + 104 x^{3} + 108 x^{2} + 48 x + 8$