Properties

Label 7.12.11.3
Base \(\Q_{7}\)
Degree \(12\)
e \(12\)
f \(1\)
c \(11\)
Galois group $D_4 \times C_3$ (as 12T14)

Related objects

Learn more about

Defining polynomial

\( x^{12} + 224 \)

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

$\Q_{7}(\sqrt{7*})$, 7.3.2.3, 7.4.3.1, 7.6.5.6

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:\( x^{12} + 224 \)

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:$x^{12} + 126 x^{8} + 2625 x^{4} + 1792$