Defining polynomial
| \( x^{14} + 4 x^{13} + 8 x^{12} + 4 x^{11} + 5 x^{10} + 8 x^{9} - 6 x^{8} - 6 x^{7} + x^{6} + 6 x^{5} + 2 x^{3} + 7 x^{2} + 6 x - 7 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $14$ |
| Ramification exponent $e$ : | $2$ |
| Residue field degree $f$ : | $7$ |
| Discriminant exponent $c$ : | $21$ |
| Discriminant root field: | $\Q_{2}(\sqrt{2})$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 2 })|$: | $14$ |
| This field is Galois and abelian over $\Q_{2}$. | |
Intermediate fields
| $\Q_{2}(\sqrt{2})$, 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} + 4 t^{3} x + 2 t^{6} + 6 t^{5} + 6 t^{4} + 2 t^{3} + 2 t^{2} + 2 t \in\Q_{2}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_{14}$ (as 14T1) |
| Inertia group: | Intransitive group isomorphic to $C_2$ |
| Unramified degree: | $7$ |
| Tame degree: | $1$ |
| Wild slopes: | [3] |
| Galois mean slope: | $3/2$ |
| Galois splitting model: | $x^{14} + 84 x^{12} + 2156 x^{10} + 19208 x^{8} + 60368 x^{6} + 47040 x^{4} + 12544 x^{2} + 896$ |