Properties

Label 7.12.11.4
Base \(\Q_{7}\)
Degree \(12\)
e \(12\)
f \(1\)
c \(11\)
Galois group $D_4 \times C_3$ (as 12T14)

Related objects

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Defining polynomial

\( x^{12} - 7 \)

Invariants

Base field: $\Q_{7}$
Degree $d$ : $12$
Ramification exponent $e$ : $12$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $11$
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

$\Q_{7}(\sqrt{7})$, 7.3.2.2, 7.4.3.2, 7.6.5.2

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{7}$
Relative Eisenstein polynomial:\( x^{12} - 7 \)

Invariants of the Galois closure

Galois group:$C_3\times D_4$ (as 12T14)
Inertia group:$C_{12}$
Unramified degree:$2$
Tame degree:$12$
Wild slopes:None
Galois mean slope:$11/12$
Galois splitting model:$x^{12} - 2 x^{11} - 4 x^{10} - 4 x^{9} - 2 x^{8} + 8 x^{7} + 7 x^{6} + 8 x^{5} - 2 x^{4} - 4 x^{3} - 4 x^{2} - 2 x + 1$