Defining polynomial
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\(x^{12} + 3 x^{3} + 6 x^{2} + 3\)
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Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $12$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $13$ |
| Discriminant root field: | $\Q_{3}(\sqrt{3})$ |
| Root number: | $i$ |
| $\card{ \Aut(K/\Q_{ 3 }) }$: | $3$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible slopes: | $[5/4]$ |
Intermediate fields
| $\Q_{3}(\sqrt{3\cdot 2})$, 3.4.3.1 |
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: |
\( x^{12} + 3 x^{3} + 6 x^{2} + 3 \)
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Ramification polygon
| Residual polynomials: | $z^{2} + 2$,$z^{9} + z^{6} + 1$ |
| Associated inertia: | $1$,$2$ |
| Indices of inseparability: | $[2, 0]$ |
Invariants of the Galois closure
| Galois group: | $S_3^2:C_6$ (as 12T121) |
| Inertia group: | $C_3^2:C_4$ (as 12T17) |
| Wild inertia group: | $C_3^2$ |
| Unramified degree: | $6$ |
| Tame degree: | $4$ |
| Wild slopes: | $[5/4, 5/4]$ |
| Galois mean slope: | $43/36$ |
| Galois splitting model: |
$x^{12} - 6 x^{11} + 18 x^{10} - 20 x^{9} + 15 x^{8} + 12 x^{7} - 18 x^{6} + 54 x^{5} + 39 x^{4} + 22 x^{3} + 39 x^{2} + 12 x + 1$
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