Defining polynomial
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\(x^{15} + 6 x^{14} + 216 x^{13} + 4272 x^{12} + 32373 x^{11} + 101574 x^{10} + 55899 x^{9} - 94851 x^{8} - 2511 x^{7} + 76572 x^{6} + 21789 x^{5} + 41553 x^{4} - 7857 x^{3} + 17496 x^{2} - 1458 x + 243\)
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Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $15$ |
| Ramification exponent $e$: | $3$ |
| Residue field degree $f$: | $5$ |
| Discriminant exponent $c$: | $15$ |
| Discriminant root field: | $\Q_{3}(\sqrt{3})$ |
| Root number: | $-i$ |
| $\card{ \Aut(K/\Q_{ 3 }) }$: | $1$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible slopes: | $[3/2]$ |
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: |
\( x^{3} + \left(3 t^{4} + 3 t^{3} + 6 t^{2} + 6\right) x^{2} + \left(6 t^{4} + 3 t^{3} + 6 t^{2} + 6 t + 6\right) x + 3 \)
$\ \in\Q_{3}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + t^{4} + 2t^{3} + t^{2} + t + 1$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_7^3:C_6$ (as 15T44) |
| Inertia group: | Intransitive group isomorphic to $C_3^4:S_3$ |
| Wild inertia group: | $C_3^5$ |
| Unramified degree: | $5$ |
| Tame degree: | $2$ |
| Wild slopes: | $[3/2, 3/2, 3/2, 3/2, 3/2]$ |
| Galois mean slope: | $727/486$ |
| Galois splitting model: |
$x^{15} - 3 x^{14} + 1860 x^{13} + 15963 x^{12} + 896346 x^{11} + 16721277 x^{10} + 130568496 x^{9} + 3483291876 x^{8} + 19775954115 x^{7} - 99312068390 x^{6} + 1374514150824 x^{5} - 20592035248785 x^{4} - 343708274699123 x^{3} + 1759644758039379 x^{2} + 11997400504345188 x - 50703945932444881$
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