Defining polynomial
|
\(x^{15} + 85 x^{12} + 3 x^{11} + 42 x^{10} + 2890 x^{9} - 765 x^{8} - 21417 x^{7} + 49214 x^{6} - 4614 x^{5} + 546465 x^{4} + 424746 x^{3} + 247392 x^{2} - 1027674 x + 1410480\)
|
Invariants
| Base field: | $\Q_{17}$ |
| Degree $d$: | $15$ |
| Ramification exponent $e$: | $3$ |
| Residue field degree $f$: | $5$ |
| Discriminant exponent $c$: | $10$ |
| Discriminant root field: | $\Q_{17}(\sqrt{3})$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 17 }) }$: | $5$ |
| This field is not Galois over $\Q_{17}.$ | |
| Visible slopes: | None |
Intermediate fields
| 17.3.2.1, 17.5.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 17.5.0.1 $\cong \Q_{17}(t)$ where $t$ is a root of
\( x^{5} + x + 14 \)
|
| Relative Eisenstein polynomial: |
\( x^{3} + 17 \)
$\ \in\Q_{17}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^{2} + 3z + 3$ |
| Associated inertia: | $2$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_5\times S_3$ (as 15T4) |
| Inertia group: | Intransitive group isomorphic to $C_3$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $10$ |
| Tame degree: | $3$ |
| Wild slopes: | None |
| Galois mean slope: | $2/3$ |
| Galois splitting model: | Not computed |