Defining polynomial
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\(x^{12} + 6 x^{11} + 24 x^{10} + 42 x^{9} + 36 x^{8} + 18 x^{7} - 15 x^{6} + 54 x^{5} + 54 x^{4} - 72 x^{3} + 585\)
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Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $6$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $18$ |
| 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
| $\Q_{3}(\sqrt{2})$, 3.4.2.2 |
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: |
\( x^{6} + \left(3 t + 6\right) x^{5} + \left(3 t + 6\right) x^{4} + 3 x^{3} + 21 t + 9 \)
$\ \in\Q_{3}(t)[x]$
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Ramification polygon
| Residual polynomials: | $2z^{2} + t + 1$,$z^{3} + 2$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[4, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_3^2:C_{12}$ (as 12T73) |
| Inertia group: | Intransitive group isomorphic to $C_3^2:C_6$ |
| Wild inertia group: | $C_3^3$ |
| Unramified degree: | $2$ |
| Tame degree: | $2$ |
| Wild slopes: | $[3/2, 3/2, 2]$ |
| Galois mean slope: | $97/54$ |
| Galois splitting model: |
$x^{12} - 84 x^{10} - 140 x^{9} + 4158 x^{8} + 252 x^{7} - 84154 x^{6} - 15876 x^{5} + 1566285 x^{4} - 3091312 x^{3} + 3313674 x^{2} - 30417828 x + 74869207$
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