Defining polynomial
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\(x^{9} + 18 x^{8} + 9 x^{7} + 9 x^{3} + 3\)
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Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $9$ |
| Ramification index $e$: | $9$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $24$ |
| 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: | $[\frac{5}{2}, \frac{19}{6}]$ |
| Visible Swan slopes: | $[\frac{3}{2},\frac{13}{6}]$ |
| Means: | $\langle1, \frac{16}{9}\rangle$ |
| Rams: | $(\frac{3}{2}, \frac{7}{2})$ |
| Jump set: | undefined |
| Roots of unity: | $2 = (3 - 1)$ |
Intermediate fields
| 3.1.3.5a1.2 |
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: |
\( x^{9} + 18 x^{8} + 9 x^{7} + 9 x^{3} + 3 \)
|
Ramification polygon
| Residual polynomials: | $z^3 + 2$,$2 z + 1$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[16, 9, 0]$ |
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: | $[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}, \frac{19}{6}]$ |
| Galois Swan slopes: | $[\frac{1}{2},\frac{3}{2},\frac{5}{3},\frac{13}{6}]$ |
| Galois mean slope: | $2.932098765432099$ |
| Galois splitting model: | $x^{9} - 3 x^{6} + 81 x^{5} - 162 x^{4} + 48 x^{3} + 729 x^{2} - 1350 x + 1163$ |