Defining polynomial
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\(x^{12} + 6 x^{11} + 9 x^{10} - 36 x^{7} - 105 x^{6} + 90 x^{5} + 432 x^{4} + 774 x^{3} + 1404 x^{2} + 864 x + 450\)
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Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $6$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $20$ |
| Discriminant root field: | $\Q_{3}$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 3 }) }$: | $1$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible slopes: | $[9/4]$ |
Intermediate fields
| $\Q_{3}(\sqrt{2})$, 3.4.2.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{3}(\sqrt{2})$ $\cong \Q_{3}(t)$ where $t$ is a root of
\( x^{2} + 2 x + 2 \)
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| Relative Eisenstein polynomial: |
\( x^{6} + 3 x^{5} + \left(3 t + 3\right) x^{3} + \left(18 t + 18\right) x^{2} + 18 t x + 21 t + 18 \)
$\ \in\Q_{3}(t)[x]$
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Ramification polygon
| Residual polynomials: | $2z + t + 2$,$z^{3} + 2$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[5, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_3^4:\OD_{16}$ (as 12T215) |
| Inertia group: | Intransitive group isomorphic to $C_3^4:C_4$ |
| Wild inertia group: | $C_3^4$ |
| Unramified degree: | $4$ |
| Tame degree: | $4$ |
| Wild slopes: | $[9/4, 9/4, 9/4, 9/4]$ |
| Galois mean slope: | $241/108$ |
| Galois splitting model: |
$x^{12} - 3 x^{10} - 16 x^{9} - 81 x^{8} + 36 x^{7} + 9 x^{6} - 162 x^{5} + 297 x^{4} + 332 x^{3} - 216 x^{2} - 168 x + 16$
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