Defining polynomial
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\(x^{15} + 10 x^{13} + 5 x^{12} + 40 x^{11} + 49 x^{10} + 90 x^{9} + 30 x^{8} - 130 x^{7} + 410 x^{6} + 269 x^{5} + 765 x^{4} - 515 x^{3} + 730 x^{2} + 205 x + 94\)
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Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $15$ |
| Ramification exponent $e$: | $5$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $12$ |
| Discriminant root field: | $\Q_{3}(\sqrt{2})$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 3 }) }$: | $3$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible slopes: | None |
Intermediate fields
| 3.3.0.1, 3.5.4.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 3.3.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of
\( x^{3} + 2 x + 1 \)
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| Relative Eisenstein polynomial: |
\( x^{5} + 3 \)
$\ \in\Q_{3}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^{4} + 2z^{3} + z^{2} + z + 2$ |
| Associated inertia: | $4$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_3\times F_5$ (as 15T8) |
| Inertia group: | Intransitive group isomorphic to $C_5$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $12$ |
| Tame degree: | $5$ |
| Wild slopes: | None |
| Galois mean slope: | $4/5$ |
| Galois splitting model: |
$x^{15} - 48 x^{10} - 513 x^{5} + 27$
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