Properties

Label 7.6.5.6
Base \(\Q_{7}\)
Degree \(6\)
e \(6\)
f \(1\)
c \(5\)
Galois group $C_6$ (as 6T1)

Related objects

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Defining polynomial

\( x^{6} + 224 \)

Invariants

Base field: $\Q_{7}$
Degree $d$ : $6$
Ramification exponent $e$ : $6$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $5$
Discriminant root field: $\Q_{7}(\sqrt{7*})$
Root number: $-i$
$|\Gal(K/\Q_{ 7 })|$: $6$
This field is Galois and abelian over $\Q_{7}$.

Intermediate fields

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

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:\( x^{6} + 224 \)

Invariants of the Galois closure

Galois group:$C_6$ (as 6T1)
Inertia group:$C_6$
Unramified degree:$1$
Tame degree:$6$
Wild slopes:None
Galois mean slope:$5/6$
Galois splitting model:$x^{6} - 14 x^{3} + 63 x^{2} - 210 x + 224$