Defining polynomial
|
\(x^{22} + 2 x^{21} + 4 x^{20} + 4 x^{19} + 4 x^{17} + 4 x^{15} + 4 x^{11} + 2\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $22$ |
| Ramification index $e$: | $22$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $42$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-1})$ |
| Root number: | $i$ |
| $\Aut(K/\Q_{2})$: | $C_2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[\frac{32}{11}]$ |
| Visible Swan slopes: | $[\frac{21}{11}]$ |
| Means: | $\langle\frac{21}{22}\rangle$ |
| Rams: | $(21)$ |
| Jump set: | $[11, 33]$ |
| Roots of unity: | $2$ |
Intermediate fields
| 2.1.11.10a1.1 |
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: |
\( x^{22} + 2 x^{21} + 4 x^{20} + 4 x^{19} + 4 x^{17} + 4 x^{15} + 4 x^{11} + 2 \)
|
Ramification polygon
| Residual polynomials: | $z^{20} + z^{18} + z^{16} + z^{14} + z^4 + z^2 + 1$,$z + 1$ |
| Associated inertia: | $10$,$1$ |
| Indices of inseparability: | $[21, 0]$ |
Invariants of the Galois closure
| Galois degree: | $225280$ |
| Galois group: | $C_2\times C_2^{10}.F_{11}$ (as 22T37) |
| 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 |