Defining polynomial
|
\(x^{20} + 8 x^{13} + 18\)
|
Invariants
| Base field: | $\Q_{2}$ |
|
| Degree $d$: | $20$ |
|
| Ramification index $e$: | $20$ |
|
| Residue field degree $f$: | $1$ |
|
| Discriminant exponent $c$: | $59$ |
|
| Discriminant root field: | $\Q_{2}(\sqrt{2})$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{2})$: | $C_2$ | |
| This field is not Galois over $\Q_{2}.$ | ||
| Visible Artin slopes: | $[3, 4]$ | |
| Visible Swan slopes: | $[2,3]$ | |
| Means: | $\langle1, 2\rangle$ | |
| Rams: | $(10, 20)$ | |
| Jump set: | $[5, 15, 35]$ | |
| Roots of unity: | $2$ |
|
Intermediate fields
| $\Q_{2}(\sqrt{-2})$, 2.1.5.4a1.1, 2.1.10.19a1.1 |
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: |
\( x^{20} + 8 x^{13} + 18 \)
|
Ramification polygon
| Residual polynomials: | $z^{16} + z^{12} + 1$,$z^2 + 1$,$z + 1$ |
| Associated inertia: | $4$,$1$,$1$ |
| Indices of inseparability: | $[40, 20, 0]$ |
Invariants of the Galois closure
| Galois degree: | $2560$ |
| Galois group: | $D_4\times C_2^4:F_5$ (as 20T261) |
| 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 |