Defining polynomial
|
\(x^{20} + 2 x^{19} + 4 x^{11} + 4 x^{9} + 2 x^{4} + 2 x^{2} + 4 x + 2\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $20$ |
| Ramification index $e$: | $20$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $38$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-5})$ |
| Root number: | $i$ |
| $\Aut(K/\Q_{2})$: | $C_2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[\frac{6}{5}, \frac{14}{5}]$ |
| Visible Swan slopes: | $[\frac{1}{5},\frac{9}{5}]$ |
| Means: | $\langle\frac{1}{10}, \frac{19}{20}\rangle$ |
| Rams: | $(1, 17)$ |
| Jump set: | $[5, 11, 31]$ |
| Roots of unity: | $2$ |
Intermediate fields
| 2.1.5.4a1.1, 2.1.10.10a1.2 |
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: |
\( x^{20} + 2 x^{19} + 4 x^{11} + 4 x^{9} + 2 x^{4} + 2 x^{2} + 4 x + 2 \)
|
Ramification polygon
| Residual polynomials: | $z^{16} + z^{12} + 1$,$z^2 + 1$,$z + 1$ |
| Associated inertia: | $4$,$1$,$1$ |
| Indices of inseparability: | $[19, 2, 0]$ |
Invariants of the Galois closure
| Galois degree: | $20480$ |
| Galois group: | $C_2^9.(C_2\times F_5)$ (as 20T530) |
| Inertia group: | not computed |
| Wild inertia group: | not computed |
| Galois unramified degree: | not computed |
| Galois tame degree: | not computed |
| Galois Artin slopes: | not computed |
| Galois Swan slopes: | not computed |
| Galois mean slope: | not computed |
| Galois splitting model: | not computed |