Defining polynomial
| \( x^{14} - 2 x^{12} - 2 x^{11} + 2 x^{10} - 2 x^{8} - 2 x^{7} + 2 x^{6} + 4 x^{5} + 2 x^{4} + 4 x^{3} + 2 x^{2} - 2 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $14$ |
| Ramification exponent $e$ : | $14$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $20$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-*})$ |
| Root number: | $-i$ |
| $|\Aut(K/\Q_{ 2 })|$: | $2$ |
| 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: | \( x^{14} - 2 x^{12} - 2 x^{11} + 2 x^{10} - 2 x^{8} - 2 x^{7} + 2 x^{6} + 4 x^{5} + 2 x^{4} + 4 x^{3} + 2 x^{2} - 2 \) |
Invariants of the Galois closure
| Galois group: | 14T44 |
| Inertia group: | $C_2 \wr C_7$ |
| Unramified degree: | $3$ |
| Tame degree: | $7$ |
| Wild slopes: | [8/7, 8/7, 8/7, 12/7, 12/7, 12/7, 2] |
| Galois mean slope: | $815/448$ |
| Galois splitting model: | $x^{14} - 14 x^{12} - 112 x^{10} + 336 x^{8} - 1704 x^{7} + 4368 x^{6} - 8736 x^{5} + 15008 x^{4} - 8736 x^{3} - 4928 x^{2} + 1344 x + 96$ |