Defining polynomial
| \( x^{14} + 2 x^{13} + 4 x^{12} + 4 x^{11} + 2 x^{10} + 2 x^{9} + 4 x^{8} + 4 x^{6} + 4 x^{5} + 2 x^{4} + 2 x^{2} + 4 x - 2 \) |
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: | \( x^{14} + 2 x^{13} + 4 x^{12} + 4 x^{11} + 2 x^{10} + 2 x^{9} + 4 x^{8} + 4 x^{6} + 4 x^{5} + 2 x^{4} + 2 x^{2} + 4 x - 2 \) |
Invariants of the Galois closure
| Galois group: | 14T35 |
| Inertia group: | 14T21 |
| Unramified degree: | $3$ |
| Tame degree: | $7$ |
| Wild slopes: | [12/7, 12/7, 12/7, 16/7, 16/7, 16/7] |
| Galois mean slope: | $493/224$ |
| Galois splitting model: | $x^{14} - 28 x^{12} - 14 x^{11} + 238 x^{10} + 238 x^{9} - 714 x^{8} - 1256 x^{7} + 644 x^{6} + 2912 x^{5} + 448 x^{4} - 2744 x^{3} - 686 x^{2} + 784 x + 210$ |