Defining polynomial
| \( x^{10} - 5 x^{8} + 14 x^{6} - 22 x^{4} + 17 x^{2} - 37 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $10$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $5$ |
| Discriminant exponent $c$: | $10$ |
| Discriminant root field: | $\Q_{2}(\sqrt{5})$ |
| Root number: | $-1$ |
| $|\Aut(K/\Q_{ 2 })|$: | $2$ |
| This field is not Galois over $\Q_{2}.$ | |
Intermediate fields
| 2.5.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 2.5.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{5} + x^{2} + 1 \) |
| Relative Eisenstein polynomial: | $ x^{2} + 2 x - 2 t^{4} - 2 t \in\Q_{2}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_2\times C_2^4:C_5$ (as 10T14) |
| Inertia group: | Intransitive group isomorphic to $C_2^4$ |
| Unramified degree: | $10$ |
| Tame degree: | $1$ |
| Wild slopes: | [2, 2, 2, 2] |
| Galois mean slope: | $15/8$ |
| Galois splitting model: | $x^{10} - 11 x^{6} - 11 x^{4} + 11 x^{2} + 11$ |