Defining polynomial
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\(x^{15} - 18 x^{14} + 174 x^{13} + 348 x^{12} - 9234 x^{11} + 11889 x^{10} + 136521 x^{9} + 196020 x^{8} + 62451 x^{7} - 5157 x^{6} + 3645 x^{5} - 32238 x^{4} - 16605 x^{3} + 13851 x^{2} - 1944 x + 243\)
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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: |
\( x^{3} + \left(6 t^{4} + 3 t^{2} + 3 t + 6\right) x^{2} + \left(3 t^{4} + 3 t^{3} + 6 t^{2} + 3 t\right) x + 3 \)
$\ \in\Q_{3}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + 2t^{4} + 2t^{3} + t^{2} + 2t$ |
| 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: |
$x^{15} - 3 x^{14} - 1077 x^{13} + 34208 x^{12} + 77190 x^{11} - 26398689 x^{10} + 420409634 x^{9} + 3065112471 x^{8} - 151805442603 x^{7} + 1553331870390 x^{6} - 1051227957054 x^{5} - 170431092049482 x^{4} + 3242321785517978 x^{3} - 36302369866746282 x^{2} + 220224351887218176 x - 524639048971169441$
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