Defining polynomial
| \( x^{14} + 4 x^{13} - 2 x^{8} + 2 x^{6} + 4 x^{5} + 4 x^{4} + 4 x^{3} + 4 x - 2 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $14$ |
| Ramification exponent $e$ : | $14$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $27$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-2*})$ |
| 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: | \( x^{14} + 4 x^{13} - 2 x^{8} + 2 x^{6} + 4 x^{5} + 4 x^{4} + 4 x^{3} + 4 x - 2 \) |
Invariants of the Galois closure
| Galois group: | 14T44 |
| Inertia group: | $C_2 \wr C_7$ |
| Unramified degree: | $3$ |
| Tame degree: | $7$ |
| Wild slopes: | [18/7, 18/7, 18/7, 20/7, 20/7, 20/7, 3] |
| Galois mean slope: | $649/224$ |
| Galois splitting model: | $x^{14} + 70 x^{12} - 2450 x^{10} - 134750 x^{8} + 2633750 x^{6} + 306250 x^{4} - 10718750 x^{2} + 7656250$ |