Defining polynomial
| \( x^{8} + 2 x^{7} + 8 x^{2} + 48 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $8$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $4$ |
| Discriminant exponent $c$: | $8$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $-1$ |
| $|\Aut(K/\Q_{ 2 })|$: | $4$ |
| This field is not Galois over $\Q_{2}.$ | |
Intermediate fields
| $\Q_{2}(\sqrt{5})$, 2.4.0.1, 2.4.4.3, 2.4.4.4 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 2.4.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{4} - x + 1 \) |
| Relative Eisenstein polynomial: | $ x^{2} + \left(6 t^{3} + 4 t^{2} + 2 t\right) x + 4 t^{3} + 2 t^{2} + 6 \in\Q_{2}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_2^2:C_4$ (as 8T10) |
| Inertia group: | Intransitive group isomorphic to $C_2^2$ |
| Unramified degree: | $4$ |
| Tame degree: | $1$ |
| Wild slopes: | [2, 2] |
| Galois mean slope: | $3/2$ |
| Galois splitting model: | $x^{8} - 2 x^{6} + 4 x^{4} - 3 x^{2} + 1$ |