Defining polynomial
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\(x^{12} - 36 x^{10} + 12 x^{9} + 540 x^{8} - 324 x^{7} - 3402 x^{6} + 3240 x^{5} + 8424 x^{4} - 10260 x^{3} + 4860 x^{2} - 972 x + 81\)
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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: |
\( x^{3} + \left(6 t^{3} + 3 t^{2}\right) x + 3 \)
$\ \in\Q_{3}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + t^{3} + 2t^{2}$ |
| 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: |
$x^{12} + 3 x^{10} - 13 x^{9} - 54 x^{8} - 42 x^{7} + 30 x^{6} + 198 x^{5} + 192 x^{4} - 125 x^{3} - 207 x^{2} - 48 x + 52$
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