Defining polynomial
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\(x^{9} + 6 x^{6} + 18 x^{3} + 9 x^{2} + 9 x + 21\)
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Invariants
| Base field: | $\Q_{3}$ |
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| Degree $d$: | $9$ |
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| Ramification index $e$: | $9$ |
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| Residue field degree $f$: | $1$ |
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| Discriminant exponent $c$: | $18$ |
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| Discriminant root field: | $\Q_{3}$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{3})$: | $C_3$ | |
| This field is not Galois over $\Q_{3}.$ | ||
| Visible Artin slopes: | $[2, \frac{7}{3}]$ | |
| Visible Swan slopes: | $[1,\frac{4}{3}]$ | |
| Means: | $\langle\frac{2}{3}, \frac{10}{9}\rangle$ | |
| Rams: | $(1, 2)$ | |
| Jump set: | undefined | |
| Roots of unity: | $2 = (3 - 1)$ |
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Intermediate fields
| 3.1.3.4a2.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{3}$ |
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| Relative Eisenstein polynomial: |
\( x^{9} + 6 x^{6} + 18 x^{3} + 9 x^{2} + 9 x + 21 \)
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Ramification polygon
| Residual polynomials: | $z^6 + 2$,$2 z^2 + 1$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[10, 6, 0]$ |
Invariants of the Galois closure
| Galois degree: | $81$ |
| Galois group: | $C_3\wr C_3$ (as 9T17) |
| Inertia group: | $\He_3$ (as 9T7) |
| Wild inertia group: | $\He_3$ |
| Galois unramified degree: | $3$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[2, 2, \frac{7}{3}]$ |
| Galois Swan slopes: | $[1,1,\frac{4}{3}]$ |
| Galois mean slope: | $2.1481481481481484$ |
| Galois splitting model: | $x^{9} - 144 x^{7} - 624 x^{6} + 1998 x^{5} + 11583 x^{4} - 8100 x^{3} - 62595 x^{2} + 16497 x + 86437$ |