Defining polynomial
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\(x^{12} + 6 x^{11} + 9 x^{4} + 18 x^{2} + 9 x + 3\)
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Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $12$ |
| Ramification index $e$: | $12$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $22$ |
| Discriminant root field: | $\Q_{3}$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{3})$: | $C_1$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible Artin slopes: | $[\frac{19}{8}]$ |
| Visible Swan slopes: | $[\frac{11}{8}]$ |
| Means: | $\langle\frac{11}{12}\rangle$ |
| Rams: | $(\frac{11}{2})$ |
| Jump set: | $[2, 14]$ |
| Roots of unity: | $6 = (3 - 1) \cdot 3$ |
Intermediate fields
| $\Q_{3}(\sqrt{3\cdot 2})$, 3.1.4.3a1.1 |
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: |
\( x^{12} + 6 x^{11} + 9 x^{4} + 18 x^{2} + 9 x + 3 \)
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Ramification polygon
| Residual polynomials: | $z^9 + z^6 + 1$,$z + 2$ |
| Associated inertia: | $2$,$1$ |
| Indices of inseparability: | $[11, 0]$ |
Invariants of the Galois closure
| Galois degree: | $1296$ |
| Galois group: | $C_3^4:\SD_{16}$ (as 12T212) |
| Inertia group: | $C_3^2:F_9$ (as 12T173) |
| Wild inertia group: | $C_3^4$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $8$ |
| Galois Artin slopes: | $[\frac{15}{8}, \frac{15}{8}, \frac{19}{8}, \frac{19}{8}]$ |
| Galois Swan slopes: | $[\frac{7}{8},\frac{7}{8},\frac{11}{8},\frac{11}{8}]$ |
| Galois mean slope: | $2.307098765432099$ |
| Galois splitting model: |
$x^{12} - 12 x^{10} - 16 x^{9} - 54 x^{8} + 564 x^{6} + 1296 x^{5} + 2457 x^{4} + 4256 x^{3} + 3240 x^{2} + 1920 x + 400$
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