Defining polynomial
| \( x^{14} - 2 x^{11} - 2 x^{10} - 2 x^{9} - 2 x^{8} + 4 x^{7} - 2 x^{4} + 4 x^{3} + 4 x^{2} + 2 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $14$ |
| Ramification exponent $e$ : | $14$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $22$ |
| 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: | \( x^{14} - 2 x^{11} - 2 x^{10} - 2 x^{9} - 2 x^{8} + 4 x^{7} - 2 x^{4} + 4 x^{3} + 4 x^{2} + 2 \) |
Invariants of the Galois closure
| Galois group: | 14T35 |
| Inertia group: | 14T21 |
| Unramified degree: | $3$ |
| Tame degree: | $7$ |
| Wild slopes: | [10/7, 10/7, 10/7, 16/7, 16/7, 16/7] |
| Galois mean slope: | $243/112$ |
| Galois splitting model: | $x^{14} - 7 x^{12} + 21 x^{10} - 70 x^{9} - 287 x^{8} + 1340 x^{7} + 77 x^{6} - 2632 x^{5} - 6811 x^{4} + 19936 x^{3} - 10535 x^{2} - 3290 x + 1507$ |