Defining polynomial
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\(x^{12} + 3 x^{10} + 9 x^{2} + 9 x + 6\)
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Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $12$ |
| Ramification index $e$: | $12$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $21$ |
| Discriminant root field: | $\Q_{3}(\sqrt{3\cdot 2})$ |
| Root number: | $i$ |
| $\Aut(K/\Q_{3})$: | $C_1$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible Artin slopes: | $[\frac{9}{4}]$ |
| Visible Swan slopes: | $[\frac{5}{4}]$ |
| Means: | $\langle\frac{5}{6}\rangle$ |
| Rams: | $(5)$ |
| Jump set: | undefined |
| Roots of unity: | $2 = (3 - 1)$ |
Intermediate fields
| $\Q_{3}(\sqrt{3})$, 3.1.4.3a1.2 |
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: |
\( x^{12} + 3 x^{10} + 9 x^{2} + 9 x + 6 \)
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Ramification polygon
| Residual polynomials: | $z^9 + z^6 + 1$,$z^2 + 1$ |
| Associated inertia: | $2$,$2$ |
| Indices of inseparability: | $[10, 0]$ |
Invariants of the Galois closure
| Galois degree: | $216$ |
| Galois group: | $S_3^2:S_3$ (as 12T120) |
| Inertia group: | $C_3^3:C_4$ (as 12T72) |
| Wild inertia group: | $C_3^3$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $4$ |
| Galois Artin slopes: | $[\frac{3}{2}, \frac{9}{4}, \frac{9}{4}]$ |
| Galois Swan slopes: | $[\frac{1}{2},\frac{5}{4},\frac{5}{4}]$ |
| Galois mean slope: | $2.138888888888889$ |
| Galois splitting model: |
$x^{12} + 36 x^{10} - 30 x^{9} + 486 x^{8} - 810 x^{7} + 3222 x^{6} - 7290 x^{5} + 12069 x^{4} - 24030 x^{3} + 24786 x^{2} - 19440 x + 13500$
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