Defining polynomial
\(x^{14} + 28 x^{9} - 154 x^{8} + 14 x^{7} + 196 x^{4} - 2156 x^{3} - 1225 x^{2} - 1078 x + 49\) |
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 \( x^{2} + 6 x + 3 \) |
Relative Eisenstein polynomial: | \( x^{7} + 14 x^{2} + \left(35 t + 28\right) x + 7 \) $\ \in\Q_{7}(t)[x]$ |
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: | $x^{14} - 14 x^{12} + 77 x^{10} - 210 x^{8} - 176 x^{7} + 294 x^{6} + 1232 x^{5} - 196 x^{4} - 2464 x^{3} + 49 x^{2} + 1232 x + 524$ |