Defining polynomial
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\(x^{12} + 3 x^{11} + 3 x^{10} + 3 x^{6} + 18 x^{3} + 18 x + 24\)
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Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $12$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $21$ |
| 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: | $[9/4]$ |
Intermediate fields
| $\Q_{3}(\sqrt{3})$, 3.4.3.2 |
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: |
\( x^{12} + 3 x^{11} + 3 x^{10} + 3 x^{6} + 18 x^{3} + 18 x + 24 \)
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Ramification polygon
| Residual polynomials: | $z^{2} + 1$,$z^{9} + z^{6} + 1$ |
| Associated inertia: | $2$,$2$ |
| Indices of inseparability: | $[10, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_3^3:D_{12}$ (as 12T169) |
| Inertia group: | $C_3\wr C_4$ (as 12T131) |
| Wild inertia group: | $C_3^4$ |
| Unramified degree: | $2$ |
| Tame degree: | $4$ |
| Wild slopes: | $[3/2, 2, 9/4, 9/4]$ |
| Galois mean slope: | $79/36$ |
| Galois splitting model: |
$x^{12} - 8 x^{9} + 12 x^{6} - 8 x^{3} + 4$
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