Defining polynomial
|
\(x^{14} + 51\)
|
Invariants
| Base field: | $\Q_{17}$ |
| Degree $d$: | $14$ |
| Ramification index $e$: | $14$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $13$ |
| Discriminant root field: | $\Q_{17}(\sqrt{17\cdot 3})$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{17})$: | $C_2$ |
| This field is not Galois over $\Q_{17}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | undefined |
| Roots of unity: | $16 = (17 - 1)$ |
Intermediate fields
| $\Q_{17}(\sqrt{17\cdot 3})$, 17.1.7.6a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{17}$ |
| Relative Eisenstein polynomial: |
\( x^{14} + 51 \)
|
Ramification polygon
| Residual polynomials: | $z^{13} + 14 z^{12} + 6 z^{11} + 7 z^{10} + 15 z^{9} + 13 z^{8} + 11 z^{7} + 15 z^{6} + 11 z^{5} + 13 z^{4} + 15 z^{3} + 7 z^{2} + 6 z + 14$ |
| Associated inertia: | $6$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $84$ |
| Galois group: | $C_2\times F_7$ (as 14T7) |
| Inertia group: | $C_{14}$ (as 14T1) |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $6$ |
| Galois tame degree: | $14$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[]$ |
| Galois mean slope: | $0.9285714285714286$ |
| Galois splitting model: | not computed |