Defining polynomial
|
\(x^{9} + 54 x + 3\)
|
Invariants
| Base field: | $\Q_{3}$ |
|
| Degree $d$: | $9$ |
|
| Ramification index $e$: | $9$ |
|
| Residue field degree $f$: | $1$ |
|
| Discriminant exponent $c$: | $26$ |
|
| Discriminant root field: | $\Q_{3}$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{3})$: | $C_1$ | |
| This field is not Galois over $\Q_{3}.$ | ||
| Visible Artin slopes: | $[\frac{5}{2}, \frac{7}{2}]$ | |
| Visible Swan slopes: | $[\frac{3}{2},\frac{5}{2}]$ | |
| Means: | $\langle1, 2\rangle$ | |
| Rams: | $(\frac{3}{2}, \frac{9}{2})$ | |
| Jump set: | undefined | |
| Roots of unity: | $2 = (3 - 1)$ |
|
Intermediate fields
| 3.1.3.5a1.1 |
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: |
\( x^{9} + 54 x + 3 \)
|
Ramification polygon
| Residual polynomials: | $z^3 + 1$,$z + 1$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[18, 9, 0]$ |
Invariants of the Galois closure
| Galois degree: | $162$ |
| Galois group: | $C_3^3:S_3$ (as 9T21) |
| Inertia group: | $C_3^3:S_3$ (as 9T21) |
| Wild inertia group: | $C_3\wr C_3$ |
| Galois unramified degree: | $1$ |
| Galois tame degree: | $2$ |
| Galois Artin slopes: | $[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}, \frac{7}{2}]$ |
| Galois Swan slopes: | $[\frac{1}{2},\frac{3}{2},\frac{5}{3},\frac{5}{2}]$ |
| Galois mean slope: | $3.154320987654321$ |
| Galois splitting model: | $x^{9} - 18 x^{6} + 36 x^{3} - 48$ |