Defining polynomial
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\(x^{15} - 9 x^{14} + 342 x^{13} + 6387 x^{12} + 34830 x^{11} + 66663 x^{10} + 40347 x^{9} + 83835 x^{8} + 114453 x^{7} + 629154 x^{6} + 182007 x^{5} + 1585332 x^{4} + 376569 x^{3} + 2510676 x^{2} + 1235898\)
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Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $15$ |
| Ramification exponent $e$: | $3$ |
| Residue field degree $f$: | $5$ |
| Discriminant exponent $c$: | $20$ |
| Discriminant root field: | $\Q_{3}(\sqrt{2})$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 3 }) }$: | $1$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible slopes: | $[2]$ |
Intermediate fields
| 3.5.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 3.5.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of
\( x^{5} + 2 x + 1 \)
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| Relative Eisenstein polynomial: |
\( x^{3} + \left(3 t^{4} + 6 t^{3} + 6 t^{2} + 3\right) x^{2} + 18 t^{2} + 9 t + 3 \)
$\ \in\Q_{3}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^{2} + t^{4} + 2t^{3} + 2t^{2} + 1$ |
| Associated inertia: | $2$ |
| Indices of inseparability: | $[2, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_3^4:C_{10}$ (as 15T33) |
| Inertia group: | Intransitive group isomorphic to $C_3^4$ |
| Wild inertia group: | $C_3^4$ |
| Unramified degree: | $10$ |
| Tame degree: | $1$ |
| Wild slopes: | $[2, 2, 2, 2]$ |
| Galois mean slope: | $160/81$ |
| Galois splitting model: |
$x^{15} + 6 x^{13} - 8 x^{12} - 45 x^{11} - 78 x^{10} - 222 x^{9} - 576 x^{8} - 1020 x^{7} + 1458 x^{6} - 1539 x^{5} - 7194 x^{4} + 9728 x^{3} + 4968 x^{2} - 11760 x + 4192$
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