Defining polynomial
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\(x^{15} + 6 x^{11} + 3 x^{10} + 3 x^{8} + 3\)
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Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $15$ |
| Ramification index $e$: | $15$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $22$ |
| Discriminant root field: | $\Q_{3}(\sqrt{2})$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{3})$: | $C_3$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible Artin slopes: | $[\frac{9}{5}]$ |
| Visible Swan slopes: | $[\frac{4}{5}]$ |
| Means: | $\langle\frac{8}{15}\rangle$ |
| Rams: | $(4)$ |
| Jump set: | undefined |
| Roots of unity: | $2 = (3 - 1)$ |
Intermediate fields
| 3.1.5.4a1.1 |
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: |
\( x^{15} + 6 x^{11} + 3 x^{10} + 3 x^{8} + 3 \)
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Ramification polygon
| Residual polynomials: | $z^{12} + 2 z^9 + z^6 + z^3 + 2$,$2 z^2 + 1$ |
| Associated inertia: | $4$,$1$ |
| Indices of inseparability: | $[8, 0]$ |
Invariants of the Galois closure
| Galois degree: | $1620$ |
| Galois group: | $C_3^4:F_5$ (as 15T41) |
| Inertia group: | $C_3^4:C_5$ (as 15T26) |
| Wild inertia group: | $C_3^4$ |
| Galois unramified degree: | $4$ |
| Galois tame degree: | $5$ |
| Galois Artin slopes: | $[\frac{9}{5}, \frac{9}{5}, \frac{9}{5}, \frac{9}{5}]$ |
| Galois Swan slopes: | $[\frac{4}{5},\frac{4}{5},\frac{4}{5},\frac{4}{5}]$ |
| Galois mean slope: | $1.7876543209876545$ |
| Galois splitting model: |
$x^{15} + 105 x^{11} - 1377 x^{10} + 1995 x^{9} - 735 x^{8} - 2205 x^{7} + 47985 x^{6} - 94788 x^{5} + 62685 x^{4} - 190365 x^{3} - 6615 x^{2} + 161910 x + 109920$
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