Defining polynomial
|
\(x^{16} + 134\)
|
Invariants
| Base field: | $\Q_{67}$ |
| Degree $d$: | $16$ |
| Ramification index $e$: | $16$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $15$ |
| Discriminant root field: | $\Q_{67}(\sqrt{67\cdot 2})$ |
| Root number: | $i$ |
| $\Aut(K/\Q_{67})$: | $C_2$ |
| This field is not Galois over $\Q_{67}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | undefined |
| Roots of unity: | $66 = (67 - 1)$ |
Intermediate fields
| $\Q_{67}(\sqrt{67})$, 67.1.4.3a1.2, 67.1.8.7a1.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{67}$ |
| Relative Eisenstein polynomial: |
\( x^{16} + 134 \)
|
Ramification polygon
| Residual polynomials: | $z^{15} + 16 z^{14} + 53 z^{13} + 24 z^{12} + 11 z^{11} + 13 z^{10} + 35 z^9 + 50 z^8 + 6 z^7 + 50 z^6 + 35 z^5 + 13 z^4 + 11 z^3 + 24 z^2 + 53 z + 16$ |
| Associated inertia: | $4$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $64$ |
| Galois group: | $C_{16}:C_4$ (as 16T136) |
| Inertia group: | $C_{16}$ (as 16T1) |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $4$ |
| Galois tame degree: | $16$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.9375$ |
| Galois splitting model: | not computed |