Defining polynomial
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\(x^{18} + 9 x^{17} + 18 x^{16} + 9 x^{14} + 9 x^{13} + 6 x^{12} + 18 x^{10} + 3\)
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Invariants
| Base field: | $\Q_{3}$ |
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| Degree $d$: | $18$ |
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| Ramification index $e$: | $18$ |
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| Residue field degree $f$: | $1$ |
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| Discriminant exponent $c$: | $45$ |
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| Discriminant root field: | $\Q_{3}(\sqrt{3\cdot 2})$ | |
| Root number: | $i$ | |
| $\Aut(K/\Q_{3})$: | $C_9$ | |
| This field is not Galois over $\Q_{3}.$ | ||
| Visible Artin slopes: | $[2, 3]$ | |
| Visible Swan slopes: | $[1,2]$ | |
| Means: | $\langle\frac{2}{3}, \frac{14}{9}\rangle$ | |
| Rams: | $(2, 8)$ | |
| Jump set: | $[1, 22, 40]$ | |
| Roots of unity: | $18 = (3 - 1) \cdot 3^{ 2 }$ |
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Intermediate fields
| $\Q_{3}(\sqrt{3\cdot 2})$, 3.1.3.4a2.1, 3.1.6.9a1.1 |
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: |
\( x^{18} + 9 x^{17} + 18 x^{16} + 9 x^{14} + 9 x^{13} + 6 x^{12} + 18 x^{10} + 3 \)
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Ramification polygon
| Residual polynomials: | $z^9 + 2$,$2 z^6 + 1$,$z^2 + 2$ |
| Associated inertia: | $1$,$1$,$1$ |
| Indices of inseparability: | $[28, 12, 0]$ |
Invariants of the Galois closure
| Galois degree: | $54$ |
| Galois group: | $S_3\times C_9$ (as 18T16) |
| 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 |