Properties

Label 7.14.14.5
Base \(\Q_{7}\)
Degree \(14\)
e \(7\)
f \(2\)
c \(14\)
Galois group $F_7 \times C_2$ (as 14T7)

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Defining polynomial

\(x^{14} - 56 x^{9} + 56 x^{8} + 14 x^{7} - 392 x^{4} - 1568 x^{3} + 392 x^{2} + 392 x + 49\) Copy content Toggle raw display

Invariants

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

Intermediate fields

$\Q_{7}(\sqrt{3})$, 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:$\Q_{7}(\sqrt{3})$ $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{2} + 6 x + 3 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{7} + \left(14 t + 14\right) x^{2} + 28 x + 7 \) $\ \in\Q_{7}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z + 3$
Associated inertia:$1$
Indices of inseparability:$[1, 0]$

Invariants of the Galois closure

Galois group:$C_2\times F_7$ (as 14T7)
Inertia group:Intransitive group isomorphic to $F_7$
Wild inertia group:$C_7$
Unramified degree:$2$
Tame degree:$6$
Wild slopes:$[7/6]$
Galois mean slope:$47/42$
Galois splitting model: $x^{14} + 42 x^{12} + 693 x^{10} + 9856 x^{8} - 26460 x^{6} + 41328 x^{4} - 44352 x^{2} + 20736$ Copy content Toggle raw display