Properties

Label 7.14.14.13
Base \(\Q_{7}\)
Degree \(14\)
e \(7\)
f \(2\)
c \(14\)
Galois group $C_7^2:C_{12}$ (as 14T23)

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Defining polynomial

\(x^{14} + 28 x^{9} - 154 x^{8} + 14 x^{7} + 196 x^{4} - 2156 x^{3} - 1225 x^{2} - 1078 x + 49\) 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: $\Q_{7}$
Root number: $-1$
$\card{ \Aut(K/\Q_{ 7 }) }$: $1$
This field is not Galois over $\Q_{7}.$
Visible slopes:$[7/6]$

Intermediate fields

$\Q_{7}(\sqrt{3})$

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: \( x^{7} + 14 x^{2} + \left(35 t + 28\right) x + 7 \) $\ \in\Q_{7}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z + 2t + 3$
Associated inertia:$1$
Indices of inseparability:$[1, 0]$

Invariants of the Galois closure

Galois group:$C_7^2:C_{12}$ (as 14T23)
Inertia group:Intransitive group isomorphic to $C_7:F_7$
Wild inertia group:$C_7^2$
Unramified degree:$2$
Tame degree:$6$
Wild slopes:$[7/6, 7/6]$
Galois mean slope:$341/294$
Galois splitting model:$x^{14} - 14 x^{12} + 77 x^{10} - 210 x^{8} - 176 x^{7} + 294 x^{6} + 1232 x^{5} - 196 x^{4} - 2464 x^{3} + 49 x^{2} + 1232 x + 524$