Defining polynomial
|
\(x^{16} + 127\)
|
Invariants
| Base field: | $\Q_{127}$ |
|
| Degree $d$: | $16$ |
|
| Ramification index $e$: | $16$ |
|
| Residue field degree $f$: | $1$ |
|
| Discriminant exponent $c$: | $15$ |
|
| Discriminant root field: | $\Q_{127}(\sqrt{127})$ | |
| Root number: | $i$ | |
| $\Aut(K/\Q_{127})$: | $C_2$ | |
| This field is not Galois over $\Q_{127}.$ | ||
| Visible Artin slopes: | $[\ ]$ | |
| Visible Swan slopes: | $[\ ]$ | |
| Means: | $\langle\ \rangle$ | |
| Rams: | $(\ )$ | |
| Jump set: | undefined | |
| Roots of unity: | $126 = (127 - 1)$ |
|
Intermediate fields
| $\Q_{127}(\sqrt{127\cdot 3})$, 127.1.4.3a1.1, 127.1.8.7a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{127}$ |
|
| Relative Eisenstein polynomial: |
\( x^{16} + 127 \)
|
Ramification polygon
| Residual polynomials: | $z^{15} + 16 z^{14} + 120 z^{13} + 52 z^{12} + 42 z^{11} + 50 z^{10} + 7 z^9 + 10 z^8 + 43 z^7 + 10 z^6 + 7 z^5 + 50 z^4 + 42 z^3 + 52 z^2 + 120 z + 16$ |
| Associated inertia: | $2$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $32$ |
| Galois group: | $D_{16}$ (as 16T56) |
| Inertia group: | $C_{16}$ (as 16T1) |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $16$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.9375$ |
| Galois splitting model: | not computed |