Defining polynomial
| \( x^{14} + 14 x^{12} - 21 x^{11} + 7 x^{10} - 21 x^{9} + 21 x^{8} - 7 x^{7} - 7 x^{6} + 21 x^{5} - 21 x^{4} - 7 x^{3} - 14 x^{2} - 7 \) |
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
| $\Q_{7}(\sqrt{7})$, 7.7.7.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{7}$ |
| Relative Eisenstein polynomial: | \( x^{14} + 14 x^{12} - 21 x^{11} + 7 x^{10} - 21 x^{9} + 21 x^{8} - 7 x^{7} - 7 x^{6} + 21 x^{5} - 21 x^{4} - 7 x^{3} - 14 x^{2} - 7 \) |
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: | $x^{14} - 7 x^{13} + 98 x^{12} - 497 x^{11} + 3374 x^{10} - 12481 x^{9} + 57792 x^{8} - 161463 x^{7} + 806463 x^{6} - 1901522 x^{5} + 10883453 x^{4} - 18763696 x^{3} + 74344025 x^{2} - 65255540 x + 161830783$ |