Defining polynomial
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\(x^{12} + 12 x^{9} - 6 x^{8} + 30 x^{6} - 36 x^{5} + 45 x^{4} - 36 x^{3} + 54 x^{2} + 18\)
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Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $6$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $14$ |
| 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: | $[3/2]$ |
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: |
\( x^{6} + 6 x^{3} + \left(6 t + 3\right) x^{2} + 3 t \)
$\ \in\Q_{3}(t)[x]$
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Ramification polygon
| Residual polynomials: | $2z^{2} + t + 1$,$z^{3} + 2$ |
| Associated inertia: | $1$,$1$ |
| 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 $C_3:S_3$ |
| Wild inertia group: | $C_3^2$ |
| Unramified degree: | $6$ |
| Tame degree: | $2$ |
| Wild slopes: | $[3/2, 3/2]$ |
| Galois mean slope: | $25/18$ |
| Galois splitting model: |
$x^{12} + 84 x^{10} - 196 x^{9} + 2646 x^{8} - 12348 x^{7} + 54446 x^{6} - 259308 x^{5} + 925365 x^{4} - 2600528 x^{3} + 7674282 x^{2} - 16492812 x + 19131511$
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