Defining polynomial
|
\(x^{8} + 4 x^{7} + 4 x^{5} + 4 x^{2} + 10\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $8$ |
| Ramification index $e$: | $8$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $20$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{2})$: | $C_2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[\frac{8}{3}, \frac{8}{3}, 3]$ |
| Visible Swan slopes: | $[\frac{5}{3},\frac{5}{3},2]$ |
| Means: | $\langle\frac{5}{6}, \frac{5}{4}, \frac{13}{8}\rangle$ |
| Rams: | $(\frac{5}{3}, \frac{5}{3}, 3)$ |
| Jump set: | $[1, 3, 7, 15]$ |
| Roots of unity: | $2$ |
Intermediate fields
| $\Q_{2}(\sqrt{2\cdot 5})$, 2.1.4.8a1.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{2}$ |
| Relative Eisenstein polynomial: |
\( x^{8} + 4 x^{7} + 4 x^{5} + 4 x^{2} + 10 \)
|
Ramification polygon
| Residual polynomials: | $z^2 + 1$,$z + 1$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[13, 10, 8, 0]$ |
Invariants of the Galois closure
| Galois degree: | $48$ |
| Galois group: | $C_2\times S_4$ (as 8T24) |
| Inertia group: | $C_2\times A_4$ (as 8T13) |
| Wild inertia group: | $C_2^3$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $3$ |
| Galois Artin slopes: | $[\frac{8}{3}, \frac{8}{3}, 3]$ |
| Galois Swan slopes: | $[\frac{5}{3},\frac{5}{3},2]$ |
| Galois mean slope: | $2.5833333333333335$ |
| Galois splitting model: | $x^{8} - 4 x^{7} + 4 x^{6} - 12 x^{5} + 24 x^{4} + 36 x^{2} + 22$ |