Defining polynomial
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\(x^{15} + 30 x^{14} + 360 x^{13} + 2175 x^{12} + 6840 x^{11} + 11016 x^{10} + 13050 x^{9} + 21060 x^{8} + 9720 x^{7} + 24084 x^{6} + 55728 x^{5} + 167184 x^{4} + 79137 x^{3} + 474822 x^{2} + 138024\)
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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: | $\Q_{3}$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 3 }) }$: | $15$ |
| This field is Galois and abelian over $\Q_{3}.$ | |
| Visible slopes: | $[2]$ |
Intermediate fields
| 3.3.4.2, 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: |
\( x^{3} + 6 x^{2} + 9 t^{3} + 3 \)
$\ \in\Q_{3}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^{2} + 2$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[2, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_{15}$ (as 15T1) |
| Inertia group: | Intransitive group isomorphic to $C_3$ |
| Wild inertia group: | $C_3$ |
| Unramified degree: | $5$ |
| Tame degree: | $1$ |
| Wild slopes: | $[2]$ |
| Galois mean slope: | $4/3$ |
| Galois splitting model: |
$x^{15} - 3 x^{14} - 36 x^{13} + 151 x^{12} + 306 x^{11} - 2250 x^{10} + 1258 x^{9} + 10467 x^{8} - 18066 x^{7} - 7856 x^{6} + 39705 x^{5} - 23016 x^{4} - 8527 x^{3} + 5730 x^{2} + 2244 x + 199$
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