Properties

Label 7.14.15.8
Base \(\Q_{7}\)
Degree \(14\)
e \(14\)
f \(1\)
c \(15\)
Galois group $F_7$ (as 14T4)

Related objects

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Defining polynomial

\( x^{14} - 21 x^{12} - 21 x^{11} + 21 x^{10} + 21 x^{9} - 14 x^{7} + 7 x^{6} + 7 x^{5} + 7 x^{3} - 7 x^{2} - 14 \)

Invariants

Base field: $\Q_{7}$
Degree $d$ : $14$
Ramification exponent $e$ : $14$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $15$
Discriminant root field: $\Q_{7}(\sqrt{7})$
Root number: $i$
$|\Aut(K/\Q_{ 7 })|$: $2$
This field is not Galois over $\Q_{7}$.

Intermediate fields

$\Q_{7}(\sqrt{7})$, 7.7.7.1

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:\( x^{14} - 21 x^{12} - 21 x^{11} + 21 x^{10} + 21 x^{9} - 14 x^{7} + 7 x^{6} + 7 x^{5} + 7 x^{3} - 7 x^{2} - 14 \)

Invariants of the Galois closure

Galois group:$F_7$ (as 14T4)
Inertia group:$F_7$
Unramified degree:$1$
Tame degree:$6$
Wild slopes:[7/6]
Galois mean slope:$47/42$
Galois splitting model:$x^{14} - 7 x^{13} + 28 x^{12} - 70 x^{11} + 119 x^{10} - 140 x^{9} + 105 x^{8} - 53 x^{7} + 14 x^{6} - 7 x^{5} + 42 x^{4} - 42 x^{3} + 35 x^{2} - 14 x + 4$