Defining polynomial
| \( x^{14} + 2 x^{9} + 2 x^{4} + 2 x^{2} + 2 x + 2 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $14$ |
| Ramification exponent $e$: | $14$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $14$ |
| 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: | \( x^{14} + 2 x^{9} + 2 x^{4} + 2 x^{2} + 2 x + 2 \) |
Invariants of the Galois closure
| Galois group: | $F_8:C_3$ (as 14T11) |
| Inertia group: | 14T6 |
| Unramified degree: | $3$ |
| Tame degree: | $7$ |
| Wild slopes: | [8/7, 8/7, 8/7] |
| Galois mean slope: | $31/28$ |
| Galois splitting model: | $x^{14} - 7 x^{12} - 28 x^{11} + 7 x^{10} + 168 x^{9} + 147 x^{8} - 272 x^{7} - 413 x^{6} + 364 x^{5} + 511 x^{4} - 196 x^{3} - 427 x^{2} - 420 x + 173$ |