Defining polynomial
| \( x^{14} + 4 x^{13} - x^{12} - 2 x^{11} + 2 x^{10} + 2 x^{9} + 2 x^{8} + 2 x^{7} + 2 x^{6} + 4 x^{4} - 2 x^{3} + 2 x^{2} + 2 x + 1 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $14$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $7$ |
| 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
| 2.7.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 2.7.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{7} - x + 1 \) |
| Relative Eisenstein polynomial: | $ x^{2} + \left(2 t^{5} + 2 t^{4} + 2 t^{2} + 2 t + 2\right) x + 2 t^{5} + 2 t^{4} + 2 t^{2} + 2 t \in\Q_{2}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_2^3:F_8$ (as 14T21) |
| Inertia group: | Intransitive group isomorphic to $C_2^6$ |
| Unramified degree: | $7$ |
| Tame degree: | $1$ |
| Wild slopes: | [2, 2, 2, 2, 2, 2] |
| Galois mean slope: | $63/32$ |
| Galois splitting model: | $x^{14} + 7 x^{12} - 63 x^{10} - 266 x^{8} - 245 x^{6} - 7 x^{4} + 21 x^{2} - 1$ |