Defining polynomial
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\(x^{12} - 24 x^{11} + 306 x^{10} - 2580 x^{9} + 22869 x^{8} - 7236 x^{7} + 99846 x^{6} - 4536 x^{5} + 179658 x^{4} + 3996 x^{3} + 191808 x^{2} + 110889\)
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Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $3$ |
| Residue field degree $f$: | $4$ |
| Discriminant exponent $c$: | $16$ |
| Discriminant root field: | $\Q_{3}(\sqrt{2})$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 3 }) }$: | $3$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible slopes: | $[2]$ |
Intermediate fields
| $\Q_{3}(\sqrt{2})$, 3.4.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 3.4.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of
\( x^{4} + 2 x^{3} + 2 \)
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| Relative Eisenstein polynomial: |
\( x^{3} + \left(6 t^{3} + 6 t^{2} + 6 t + 3\right) x^{2} + 18 t^{3} + 18 t^{2} + 21 \)
$\ \in\Q_{3}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^{2} + 2t^{3} + 2t^{2} + 2t + 1$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[2, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_3^2:C_{12}$ (as 12T73) |
| Inertia group: | Intransitive group isomorphic to $C_3^3$ |
| Wild inertia group: | $C_3^3$ |
| Unramified degree: | $4$ |
| Tame degree: | $1$ |
| Wild slopes: | $[2, 2, 2]$ |
| Galois mean slope: | $52/27$ |
| Galois splitting model: |
$x^{12} - 12 x^{10} - 12 x^{9} + 18 x^{8} + 36 x^{7} + 130 x^{6} + 324 x^{5} + 405 x^{4} + 432 x^{3} + 522 x^{2} + 396 x + 119$
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