Defining polynomial
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\(x^{12} - 24 x^{11} + 306 x^{10} - 2004 x^{9} + 7236 x^{8} - 4374 x^{7} - 1458 x^{6} + 5832 x^{5} - 1836 x^{3} + 324 x^{2} + 486 x + 81\)
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Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $3$ |
| Residue field degree $f$: | $4$ |
| Discriminant exponent $c$: | $12$ |
| Discriminant root field: | $\Q_{3}$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 3 }) }$: | $1$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible slopes: | $[3/2]$ |
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: |
\( x^{3} + \left(6 t^{3} + 6 t^{2}\right) x^{2} + \left(3 t + 6\right) x + 3 \)
$\ \in\Q_{3}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + 2t + 1$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_3^2:F_9$ (as 12T173) |
| Inertia group: | Intransitive group isomorphic to $C_3^3:S_3$ |
| Wild inertia group: | $C_3^4$ |
| Unramified degree: | $4$ |
| Tame degree: | $2$ |
| Wild slopes: | $[3/2, 3/2, 3/2, 3/2]$ |
| Galois mean slope: | $241/162$ |
| Galois splitting model: |
$x^{12} - 12 x^{10} - 8 x^{9} + 54 x^{8} + 72 x^{7} - 101 x^{6} - 216 x^{5} + 39 x^{4} + 286 x^{3} + 63 x^{2} - 210 x - 137$
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