Defining polynomial
|
\(x^{16} + 128 x^{15} + 7192 x^{14} + 232064 x^{13} + 4716796 x^{12} + 62185088 x^{11} + 525781480 x^{10} + 2696730752 x^{9} + 7365142088 x^{8} + 8090194432 x^{7} + 4732152320 x^{6} + 1682759680 x^{5} + 456414056 x^{4} + 996830464 x^{3} + 7439529968 x^{2} + 33582546688 x + 66368009604\)
|
Invariants
| Base field: | $\Q_{17}$ |
| Degree $d$: | $16$ |
| Ramification exponent $e$: | $8$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $14$ |
| Discriminant root field: | $\Q_{17}$ |
| Root number: | $-1$ |
| $\card{ \Gal(K/\Q_{ 17 }) }$: | $16$ |
| This field is Galois and abelian over $\Q_{17}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{17}(\sqrt{3})$, $\Q_{17}(\sqrt{17})$, $\Q_{17}(\sqrt{17\cdot 3})$, 17.4.2.1, 17.4.3.1, 17.4.3.2, 17.8.6.1, 17.8.7.1, 17.8.7.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{17}(\sqrt{3})$ $\cong \Q_{17}(t)$ where $t$ is a root of
\( x^{2} + 16 x + 3 \)
|
| Relative Eisenstein polynomial: |
\( x^{8} + 17 \)
$\ \in\Q_{17}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^{7} + 8z^{6} + 11z^{5} + 5z^{4} + 2z^{3} + 5z^{2} + 11z + 8$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_2\times C_8$ (as 16T5) |
| Inertia group: | Intransitive group isomorphic to $C_8$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $2$ |
| Tame degree: | $8$ |
| Wild slopes: | None |
| Galois mean slope: | $7/8$ |
| Galois splitting model: | Not computed |