Properties

Label 7.14.15.7
Base \(\Q_{7}\)
Degree \(14\)
e \(14\)
f \(1\)
c \(15\)
Galois group $F_7$ (as 14T4)

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Defining polynomial

\(x^{14} + 21 x^{3} + 28 x^{2} + 7\) Copy content Toggle raw display

Invariants

Base field: $\Q_{7}$
Degree $d$: $14$
Ramification exponent $e$: $14$
Residue field degree $f$: $1$
Discriminant exponent $c$: $15$
Discriminant root field: $\Q_{7}(\sqrt{7\cdot 3})$
Root number: $-i$
$\card{ \Aut(K/\Q_{ 7 }) }$: $2$
This field is not Galois over $\Q_{7}.$
Visible slopes:$[7/6]$

Intermediate fields

$\Q_{7}(\sqrt{7\cdot 3})$, 7.7.7.6

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: \( x^{14} + 21 x^{3} + 28 x^{2} + 7 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$2z^{2} + 6$,$z^{7} + 2$
Associated inertia:$1$,$1$
Indices of inseparability:$[2, 0]$

Invariants of the Galois closure

Galois group:$F_7$ (as 14T4)
Inertia group:$F_7$ (as 14T4)
Wild inertia group:$C_7$
Unramified degree:$1$
Tame degree:$6$
Wild slopes:$[7/6]$
Galois mean slope:$47/42$
Galois splitting model: $x^{14} - 28 x^{12} + 112 x^{10} + 1645 x^{8} + 5530 x^{6} + 9163 x^{4} + 7721 x^{2} + 2800$ Copy content Toggle raw display