Defining polynomial
| \( x^{14} + 4 x^{13} + x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{9} - 2 x^{8} + 4 x^{7} + 2 x^{6} - 2 x^{5} - 2 x^{4} - 2 x^{3} - 2 x^{2} + 4 x + 1 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $14$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $7$ |
| Discriminant exponent $c$: | $14$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-5})$ |
| Root number: | $i$ |
| $|\Aut(K/\Q_{ 2 })|$: | $2$ |
| 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^{6} + 2 t^{4} + 2 t^{3} + 2 t^{2} + 2\right) x + 2 t^{3} + 2 t \in\Q_{2}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_2 \wr C_7$ (as 14T29) |
| Inertia group: | Intransitive group isomorphic to $C_2^7$ |
| Unramified degree: | $7$ |
| Tame degree: | $1$ |
| Wild slopes: | [2, 2, 2, 2, 2, 2, 2] |
| Galois mean slope: | $127/64$ |
| Galois splitting model: | $x^{14} - 42 x^{12} + 567 x^{10} - 2457 x^{8} - 1701 x^{6} + 17010 x^{4} - 2187$ |