Defining polynomial
|
\(x^{14} + 7 x^{3} + 21 x^{2} + 21\)
|
Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $14$ |
| Ramification exponent $e$: | $14$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $15$ |
| Discriminant root field: | $\Q_{7}(\sqrt{7})$ |
| Root number: | $i$ |
| $\card{ \Aut(K/\Q_{ 7 }) }$: | $2$ |
| This field is not Galois over $\Q_{7}.$ | |
| Visible slopes: | $[7/6]$ |
Intermediate fields
| $\Q_{7}(\sqrt{7})$, 7.7.7.2 |
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: |
\( x^{14} + 7 x^{3} + 21 x^{2} + 21 \)
|
Ramification polygon
| Residual polynomials: | $2z^{2} + 5$,$z^{7} + 2$ |
| 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: |
$x^{14} - 21 x^{12} - 371 x^{11} - 462 x^{10} + 7014 x^{9} + 107562 x^{8} + 755091 x^{7} + 4318251 x^{6} + 17977267 x^{5} + 64337889 x^{4} + 183494808 x^{3} + 486786545 x^{2} + 638039157 x + 1697173893$
|