Properties

Label 7.6.5.5
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

\(x^{6} + 56\)  Toggle raw display

Invariants

Base field: $\Q_{7}$
Degree $d$: $6$
Ramification exponent $e$: $6$
Residue field degree $f$: $1$
Discriminant exponent $c$: $5$
Discriminant root field: $\Q_{7}(\sqrt{7\cdot 3})$
Root number: $-i$
$|\Gal(K/\Q_{ 7 })|$: $6$
This field is Galois and abelian over $\Q_{7}.$

Intermediate fields

$\Q_{7}(\sqrt{7\cdot 3})$, 7.3.2.2

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:\( x^{6} + 56 \)  Toggle raw display

Invariants of the Galois closure

Galois group:$C_6$ (as 6T1)
Inertia group:$C_6$
Wild inertia group:$C_1$
Unramified degree:$1$
Tame degree:$6$
Wild slopes:None
Galois mean slope:$5/6$
Galois splitting model:$x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$