Defining polynomial
|
\(x^{21} + 3 x^{19} + 18 x^{5} + 18 x^{4} + 3\)
|
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $21$ |
| Ramification index $e$: | $21$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $39$ |
| Discriminant root field: | $\Q_{3}(\sqrt{3})$ |
| Root number: | $i$ |
| $\Aut(K/\Q_{3})$: | $C_1$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible Artin slopes: | $[\frac{33}{14}]$ |
| Visible Swan slopes: | $[\frac{19}{14}]$ |
| Means: | $\langle\frac{19}{21}\rangle$ |
| Rams: | $(\frac{19}{2})$ |
| Jump set: | undefined |
| Roots of unity: | $2 = (3 - 1)$ |
Intermediate fields
| 3.1.7.6a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{3}$ |
| Relative Eisenstein polynomial: |
\( x^{21} + 3 x^{19} + 18 x^{5} + 18 x^{4} + 3 \)
|
Ramification polygon
| Residual polynomials: | $z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1$,$z + 2$ |
| Associated inertia: | $6$,$1$ |
| Indices of inseparability: | $[19, 0]$ |
Invariants of the Galois closure
| Galois degree: | $183708$ |
| Galois group: | $C_3^6.(S_3\times F_7)$ (as 21T107) |
| 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 |