Defining polynomial
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\(x^{15} + 6 x^{14} + 3 x^{13} + 3 x^{11} + 6 x^{10} + 12\)
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Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $15$ |
| Ramification exponent $e$: | $15$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $24$ |
| Discriminant root field: | $\Q_{3}(\sqrt{2})$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 3 }) }$: | $3$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible slopes: | $[2]$ |
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: |
\( x^{15} + 6 x^{14} + 3 x^{13} + 3 x^{11} + 6 x^{10} + 12 \)
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Ramification polygon
| Residual polynomials: | $2z^{2} + 1$,$z^{12} + 2z^{9} + z^{6} + z^{3} + 2$ |
| Associated inertia: | $1$,$4$ |
| Indices of inseparability: | $[10, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_3\wr F_5$ (as 15T56) |
| Inertia group: | $C_3\wr C_5$ (as 15T36) |
| Wild inertia group: | $C_3^5$ |
| Unramified degree: | $4$ |
| Tame degree: | $5$ |
| Wild slopes: | $[9/5, 9/5, 9/5, 9/5, 2]$ |
| Galois mean slope: | $2344/1215$ |
| Galois splitting model: |
$x^{15} - 105 x^{13} - 890 x^{12} + 9825 x^{11} + 82467 x^{10} - 711335 x^{9} - 1617705 x^{8} + 22554360 x^{7} - 56094100 x^{6} + 110309166 x^{5} - 342408585 x^{4} + 364303925 x^{3} - 520500750 x^{2} + 340121250 x - 44366875$
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