Properties

Label 7.7.12.11
Base \(\Q_{7}\)
Degree \(7\)
e \(7\)
f \(1\)
c \(12\)
Galois group $F_7$ (as 7T4)

Related objects

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Defining polynomial

\( x^{7} + 14 x^{6} + 7 \)

Invariants

Base field: $\Q_{7}$
Degree $d$ : $7$
Ramification exponent $e$ : $7$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $12$
Discriminant root field: $\Q_{7}(\sqrt{*})$
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:\( x^{7} + 14 x^{6} + 7 \)

Invariants of the Galois closure

Galois group:$F_7$ (as 7T4)
Inertia group:$C_7$
Unramified degree:$6$
Tame degree:$1$
Wild slopes:[2]
Galois mean slope:$12/7$
Galois splitting model:$x^{7} - 21 x^{5} - 70 x^{4} - 105 x^{3} + 1680 x^{2} + 10549 x + 18390$