Defining polynomial
|
\(x^{23} + 127\)
|
Invariants
| Base field: | $\Q_{127}$ |
|
| Degree $d$: | $23$ |
|
| Ramification index $e$: | $23$ |
|
| Residue field degree $f$: | $1$ |
|
| Discriminant exponent $c$: | $22$ |
|
| Discriminant root field: | $\Q_{127}$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{127})$: | $C_1$ | |
| 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
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 127 }$. |
Canonical tower
| Unramified subfield: | $\Q_{127}$ |
|
| Relative Eisenstein polynomial: |
\( x^{23} + 127 \)
|
Ramification polygon
| Residual polynomials: | $z^{22} + 23 z^{21} + 126 z^{20} + 120 z^{19} + 92 z^{18} + 121 z^{17} + 109 z^{16} + 47 z^{15} + 94 z^{14} + 72 z^{13} + 50 z^{12} + 36 z^{11} + 36 z^{10} + 50 z^9 + 72 z^8 + 94 z^7 + 47 z^6 + 109 z^5 + 121 z^4 + 92 z^3 + 120 z^2 + 126 z + 23$ |
| Associated inertia: | $11$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $253$ |
| Galois group: | $C_{23}:C_{11}$ (as 23T3) |
| Inertia group: | $C_{23}$ (as 23T1) |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $11$ |
| Galois tame degree: | $23$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.9565217391304348$ |
| Galois splitting model: | not computed |