Properties

Label 7.7.12.10
Base \(\Q_{7}\)
Degree \(7\)
e \(7\)
f \(1\)
c \(12\)
Galois group $D_{7}$ (as 7T2)

Related objects

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Defining polynomial

\( x^{7} + 7 x^{6} + 7 \)

Invariants

Base field: $\Q_{7}$
Degree $d$: $7$
Ramification exponent $e$: $7$
Residue field degree $f$: $1$
Discriminant exponent $c$: $12$
Discriminant root field: $\Q_{7}(\sqrt{3})$
Root number: $1$
$|\Aut(K/\Q_{ 7 })|$: $1$
This field is not Galois over $\Q_{7}.$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 7 }$.

Unramified/totally ramified tower

Unramified subfield:$\Q_{7}$
Relative Eisenstein polynomial:\( x^{7} + 7 x^{6} + 7 \)

Invariants of the Galois closure

Galois group:$D_7$ (as 7T2)
Inertia group:$C_7$
Unramified degree:$2$
Tame degree:$1$
Wild slopes:[2]
Galois mean slope:$12/7$
Galois splitting model:$x^{7} - 3157 x^{5} - 85526 x^{4} - 49938 x^{3} + 18478208 x^{2} + 141589728 x - 96857088$