Defining polynomial
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\(x^{15} + 9 x^{7} + 18 x^{4} + 3\)
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Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $15$ |
| Ramification index $e$: | $15$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $29$ |
| Discriminant root field: | $\Q_{3}(\sqrt{3})$ |
| Root number: | $-i$ |
| $\Aut(K/\Q_{3})$: | $C_1$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible Artin slopes: | $[\frac{5}{2}]$ |
| Visible Swan slopes: | $[\frac{3}{2}]$ |
| Means: | $\langle1\rangle$ |
| Rams: | $(\frac{15}{2})$ |
| 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: |
\( x^{15} + 9 x^{7} + 18 x^{4} + 3 \)
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Ramification polygon
| Residual polynomials: | $z^{12} + 2 z^9 + z^6 + z^3 + 2$,$2 z + 1$ |
| Associated inertia: | $4$,$1$ |
| Indices of inseparability: | $[15, 0]$ |
Invariants of the Galois closure
| Galois degree: | $9720$ |
| Galois group: | $C_3^4:(S_3\times F_5)$ (as 15T64) |
| Inertia group: | $C_3^5:C_{10}$ (as 15T44) |
| Wild inertia group: | $C_3^5$ |
| Galois unramified degree: | $4$ |
| Galois tame degree: | $10$ |
| Galois Artin slopes: | $[\frac{17}{10}, \frac{17}{10}, \frac{17}{10}, \frac{17}{10}, \frac{5}{2}]$ |
| Galois Swan slopes: | $[\frac{7}{10},\frac{7}{10},\frac{7}{10},\frac{7}{10},\frac{3}{2}]$ |
| Galois mean slope: | $2.2300411522633743$ |
| Galois splitting model: |
$x^{15} - 315 x^{12} + 35280 x^{9} - 2253510 x^{6} + 30144555 x^{3} - 132355125$
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