Defining polynomial
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\(x^{15} + 6 x^{14} + 3 x^{13} + 6 x^{9} + 9 x^{5} + 18 x^{4} + 18 x^{2} + 9 x + 21\)
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Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $15$ |
| Ramification exponent $e$: | $15$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $27$ |
| Discriminant root field: | $\Q_{3}(\sqrt{3\cdot 2})$ |
| Root number: | $i$ |
| $\card{ \Aut(K/\Q_{ 3 }) }$: | $1$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible slopes: | $[23/10]$ |
Intermediate fields
| 3.5.4.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{3}$ |
| Relative Eisenstein polynomial: |
\( x^{15} + 6 x^{14} + 3 x^{13} + 6 x^{9} + 9 x^{5} + 18 x^{4} + 18 x^{2} + 9 x + 21 \)
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Ramification polygon
| Residual polynomials: | $2z + 2$,$z^{12} + 2z^{9} + z^{6} + z^{3} + 2$ |
| Associated inertia: | $1$,$4$ |
| Indices of inseparability: | $[13, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_3^4:(S_3\times F_5)$ (as 15T64) |
| Inertia group: | $C_7^3:C_6$ (as 15T44) |
| Wild inertia group: | $C_3^5$ |
| Unramified degree: | $4$ |
| Tame degree: | $10$ |
| Wild slopes: | $[3/2, 23/10, 23/10, 23/10, 23/10]$ |
| Galois mean slope: | $1853/810$ |
| Galois splitting model: |
$x^{15} - 93 x^{13} - 277 x^{12} + 2520 x^{11} + 13020 x^{10} - 11480 x^{9} - 185220 x^{8} - 275625 x^{7} + 434140 x^{6} - 1528065 x^{5} - 20677755 x^{4} - 56521255 x^{3} - 64610910 x^{2} - 32161395 x - 6566735$
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