Defining polynomial
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\(x^{12} + 6 x^{10} + 9 x^{3} + 9 x^{2} + 6\)
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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: | $-i$ |
| $\Aut(K/\Q_{3})$: | $C_3$ |
| 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: |
\( x^{12} + 6 x^{10} + 9 x^{3} + 9 x^{2} + 6 \)
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Ramification polygon
| Residual polynomials: | $z^9 + z^6 + 1$,$z^2 + 2$ |
| Associated inertia: | $2$,$1$ |
| Indices of inseparability: | $[10, 0]$ |
Invariants of the Galois closure
| Galois degree: | $216$ |
| Galois group: | $S_3^2:C_6$ (as 12T121) |
| Inertia group: | $C_3^2:C_4$ (as 12T17) |
| Wild inertia group: | $C_3^2$ |
| Galois unramified degree: | $6$ |
| Galois tame degree: | $4$ |
| Galois Artin slopes: | $[\frac{9}{4}, \frac{9}{4}]$ |
| Galois Swan slopes: | $[\frac{5}{4},\frac{5}{4}]$ |
| Galois mean slope: | $2.0833333333333335$ |
| Galois splitting model: |
$x^{12} - 42 x^{10} - 14 x^{9} + 567 x^{8} + 504 x^{7} - 3192 x^{6} - 5292 x^{5} + 6615 x^{4} + 20188 x^{3} + 3969 x^{2} - 24108 x - 20531$
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