Defining polynomial
| \( x^{14} + 14 x^{13} + 7 x^{12} + 14 x^{11} - 7 x^{10} - 14 x^{8} + 21 x^{7} + 7 x^{6} - 7 x^{5} + 14 x^{4} - 14 x^{3} + 21 x^{2} - 7 \) |
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: | $i$ |
| $|\Aut(K/\Q_{ 7 })|$: | $2$ |
| This field is not Galois over $\Q_{7}$. | |
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: | \( x^{14} + 14 x^{13} + 7 x^{12} + 14 x^{11} - 7 x^{10} - 14 x^{8} + 21 x^{7} + 7 x^{6} - 7 x^{5} + 14 x^{4} - 14 x^{3} + 21 x^{2} - 7 \) |
Invariants of the Galois closure
| Galois group: | $F_7$ (as 14T4) |
| Inertia group: | $F_7$ |
| Unramified degree: | $1$ |
| Tame degree: | $6$ |
| Wild slopes: | [7/6] |
| Galois mean slope: | $47/42$ |
| Galois splitting model: | $x^{14} - 21 x^{12} - 371 x^{11} - 462 x^{10} + 7014 x^{9} + 107562 x^{8} + 755091 x^{7} + 4318251 x^{6} + 17977267 x^{5} + 64337889 x^{4} + 183494808 x^{3} + 486786545 x^{2} + 638039157 x + 1697173893$ |