Defining polynomial
| \( x^{14} + 14 x^{12} + 7 x^{11} + 7 x^{10} - 21 x^{9} - 21 x^{8} + 14 x^{7} - 7 x^{6} + 21 x^{5} - 21 x^{3} + 14 x^{2} - 7 \) |
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
| $\Q_{7}(\sqrt{7})$, 7.7.7.4 |
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: | \( x^{14} + 14 x^{12} + 7 x^{11} + 7 x^{10} - 21 x^{9} - 21 x^{8} + 14 x^{7} - 7 x^{6} + 21 x^{5} - 21 x^{3} + 14 x^{2} - 7 \) |
Invariants of the Galois closure
| Galois group: | $C_2\times F_7$ (as 14T7) |
| Inertia group: | $F_7$ |
| Unramified degree: | $2$ |
| Tame degree: | $6$ |
| Wild slopes: | [7/6] |
| Galois mean slope: | $47/42$ |
| Galois splitting model: | $x^{14} + 35 x^{12} + 399 x^{10} + 763 x^{8} + 476 x^{6} + 42 x^{4} - 35 x^{2} - 7$ |