Defining polynomial
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\(x^{9} + 18 x^{7} + 12 x^{6} + 27 x^{2} + 30\)
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Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $9$ |
| Ramification exponent $e$: | $9$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $23$ |
| Discriminant root field: | $\Q_{3}(\sqrt{3})$ |
| Root number: | $i$ |
| $\card{ \Aut(K/\Q_{ 3 }) }$: | $1$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible slopes: | $[2, 19/6]$ |
Intermediate fields
| 3.3.4.4 |
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: |
\( x^{9} + 18 x^{7} + 12 x^{6} + 27 x^{2} + 30 \)
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Ramification polygon
| Residual polynomials: | $z + 2$,$z^{6} + 1$ |
| Associated inertia: | $1$,$2$ |
| Indices of inseparability: | $[15, 6, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_3^3:D_6$ (as 9T24) |
| Inertia group: | $C_3^3:C_6$ (as 9T22) |
| Wild inertia group: | $C_3\wr C_3$ |
| Unramified degree: | $2$ |
| Tame degree: | $2$ |
| Wild slopes: | $[2, 5/2, 17/6, 19/6]$ |
| Galois mean slope: | $161/54$ |
| Galois splitting model: |
$x^{9} + 12 x^{6} + 66$
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