Defining polynomial
|
\(x^{14} + 14 x^{13} + 28 x^{12} + 168\)
|
Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $14$ |
| Ramification exponent $e$: | $14$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $25$ |
| Discriminant root field: | $\Q_{7}(\sqrt{7})$ |
| Root number: | $i$ |
| $\card{ \Aut(K/\Q_{ 7 }) }$: | $7$ |
| This field is not Galois over $\Q_{7}.$ | |
| Visible slopes: | $[2]$ |
Intermediate fields
| $\Q_{7}(\sqrt{7})$ |
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: |
\( x^{14} + 14 x^{13} + 28 x^{12} + 168 \)
|
Ramification polygon
| Residual polynomials: | $2z^{6} + 5$,$z^{7} + 2$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[12, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_7\times D_7$ (as 14T8) |
| Inertia group: | $C_7\times D_7$ (as 14T8) |
| Wild inertia group: | $C_7^2$ |
| Unramified degree: | $1$ |
| Tame degree: | $2$ |
| Wild slopes: | $[3/2, 2]$ |
| Galois mean slope: | $187/98$ |
| Galois splitting model: |
$x^{14} + 406 x^{12} - 2639 x^{11} + 405188 x^{10} - 2836316 x^{9} + 147585466 x^{8} - 1402725795 x^{7} + 26467222012 x^{6} - 289191009968 x^{5} + 2788835923594 x^{4} - 22815536222150 x^{3} + 178821725118890 x^{2} - 792822309773725 x + 3074089055871625$
|