Defining polynomial
\(x^{14} + 2 x^{10} + 2 x^{8} + 2 x^{7} + 2 x^{6} + 2 x^{5} + 2 x^{4} + 2 x^{3} + 2 x^{2} + 2\)  |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $14$ |
| Ramification exponent $e$: | $14$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $16$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 2 })|$: | $2$ |
| This field is not Galois over $\Q_{2}.$ | |
Intermediate fields
| 2.7.6.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{2}$ |
| Relative Eisenstein polynomial: | \( x^{14} + 2 x^{10} + 2 x^{8} + 2 x^{7} + 2 x^{6} + 2 x^{5} + 2 x^{4} + 2 x^{3} + 2 x^{2} + 2 \) |
Invariants of the Galois closure
| Galois group: | 14T35 |
| Inertia group: | 14T21 |
| Unramified degree: | $3$ |
| Tame degree: | $7$ |
| Wild slopes: | [8/7, 8/7, 8/7, 10/7, 10/7, 10/7] |
| Galois mean slope: | $311/224$ |
| Galois splitting model: | $x^{14} + 21 x^{12} - 7 x^{10} - 707 x^{8} + 2415 x^{6} - 2961 x^{4} + 1323 x^{2} - 81$ |