Defining polynomial
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\(x^{15} - 18 x^{14} + 87 x^{13} + 663 x^{12} + 7002 x^{11} + 59229 x^{10} + 24984 x^{9} + 28998 x^{8} + 14823 x^{7} + 8208 x^{6} + 729 x^{5} - 3969 x^{4} - 2997 x^{3} + 972 x^{2} + 1215 x + 243\)
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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: | $\Q_{3}(\sqrt{3\cdot 2})$ |
| Root number: | $i$ |
| $\card{ \Aut(K/\Q_{ 3 }) }$: | $5$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible slopes: | $[3/2]$ |
Intermediate fields
| 3.3.3.2, 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: |
\( x^{3} + \left(6 t^{4} + 3 t^{3} + 6\right) x^{2} + 3 x + 3 \)
$\ \in\Q_{3}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + 2$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_5\times S_3$ (as 15T4) |
| Inertia group: | Intransitive group isomorphic to $S_3$ |
| Wild inertia group: | $C_3$ |
| Unramified degree: | $5$ |
| Tame degree: | $2$ |
| Wild slopes: | $[3/2]$ |
| Galois mean slope: | $7/6$ |
| Galois splitting model: |
$x^{15} - 3 x^{14} + 18 x^{13} - 13 x^{12} + 378 x^{11} - 678 x^{10} + 2363 x^{9} + 966 x^{8} + 9105 x^{7} - 9552 x^{6} + 5115 x^{5} + 291 x^{4} - 768 x^{3} + 63 x^{2} + 18 x - 1$
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