Defining polynomial
|
\(x^{21} + 14 x^{9} + 21 x^{8} + 21\)
|
Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $21$ |
| Ramification index $e$: | $21$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $28$ |
| Discriminant root field: | $\Q_{7}(\sqrt{3})$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{7})$: | $C_1$ |
| Visible Artin slopes: | $[\frac{13}{9}]$ |
| Visible Swan slopes: | $[\frac{4}{9}]$ |
| Means: | $\langle\frac{8}{21}\rangle$ |
| Rams: | $(\frac{4}{3})$ |
| Jump set: | undefined |
| Roots of unity: | $6 = (7 - 1)$ |
Intermediate fields
| 7.1.3.2a1.3 |
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: |
\( x^{21} + 14 x^{9} + 21 x^{8} + 21 \)
|
Ramification polygon
| Residual polynomials: | $z^{14} + 3 z^7 + 3$,$3 z^2 + 6$ |
| Associated inertia: | $1$,$2$ |
| Indices of inseparability: | $[8, 0]$ |
Invariants of the Galois closure
| Galois degree: | not computed |
| Galois group: | not computed |
| 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 |