Defining polynomial
|
\(x^{19} + 7\)
|
Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $19$ |
| Ramification index $e$: | $19$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $18$ |
| Discriminant root field: | $\Q_{7}$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{7})$: | $C_1$ |
| This field is not Galois over $\Q_{7}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | undefined |
| Roots of unity: | $6 = (7 - 1)$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 7 }$. |
Canonical tower
| Unramified subfield: | $\Q_{7}$ |
| Relative Eisenstein polynomial: |
\( x^{19} + 7 \)
|
Ramification polygon
| Residual polynomials: | $z^{18} + 5 z^{17} + 3 z^{16} + 3 z^{15} + 5 z^{14} + z^{13} + 2 z^{11} + 3 z^{10} + 6 z^9 + 6 z^8 + 3 z^7 + 2 z^6 + z^4 + 5 z^3 + 3 z^2 + 3 z + 5$ |
| Associated inertia: | $3$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $57$ |
| Galois group: | $C_{19}:C_3$ (as 19T3) |
| Inertia group: | $C_{19}$ (as 19T1) |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $3$ |
| Galois tame degree: | $19$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.9473684210526315$ |
| Galois splitting model: | not computed |