Defining polynomial
|
\(x^{20} + 21\)
|
Invariants
| Base field: | $\Q_{7}$ |
|
| Degree $d$: | $20$ |
|
| Ramification index $e$: | $20$ |
|
| Residue field degree $f$: | $1$ |
|
| Discriminant exponent $c$: | $19$ |
|
| Discriminant root field: | $\Q_{7}(\sqrt{7\cdot 3})$ | |
| Root number: | $i$ | |
| $\Aut(K/\Q_{7})$: | $C_2$ | |
| This field is not Galois over $\Q_{7}.$ | ||
| Visible Artin slopes: | $[\ ]$ | |
| Visible Swan slopes: | $[\ ]$ | |
| Means: | $\langle\ \rangle$ | |
| Rams: | $(\ )$ | |
| Jump set: | undefined | |
| Roots of unity: | $6 = (7 - 1)$ |
|
Intermediate fields
| $\Q_{7}(\sqrt{7})$, 7.1.4.3a1.2, 7.1.5.4a1.1, 7.1.10.9a1.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{7}$ |
|
| Relative Eisenstein polynomial: |
\( x^{20} + 21 \)
|
Ramification polygon
| Residual polynomials: | $z^{19} + 6 z^{18} + z^{17} + 6 z^{16} + z^{15} + 6 z^{14} + z^{13} + 2 z^{12} + 5 z^{11} + 2 z^{10} + 5 z^9 + 2 z^8 + 5 z^7 + 2 z^6 + z^5 + 6 z^4 + z^3 + 6 z^2 + z + 6$ |
| Associated inertia: | $4$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $80$ |
| Galois group: | $C_{20}:C_4$ (as 20T18) |
| Inertia group: | $C_{20}$ (as 20T1) |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $4$ |
| Galois tame degree: | $20$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.95$ |
| Galois splitting model: | not computed |