Defining polynomial
|
\(x^{18} + 411\)
|
Invariants
| Base field: | $\Q_{137}$ |
|
| Degree $d$: | $18$ |
|
| Ramification index $e$: | $18$ |
|
| Residue field degree $f$: | $1$ |
|
| Discriminant exponent $c$: | $17$ |
|
| Discriminant root field: | $\Q_{137}(\sqrt{137\cdot 3})$ | |
| Root number: | $-1$ | |
| $\Aut(K/\Q_{137})$: | $C_2$ | |
| This field is not Galois over $\Q_{137}.$ | ||
| Visible Artin slopes: | $[\ ]$ | |
| Visible Swan slopes: | $[\ ]$ | |
| Means: | $\langle\ \rangle$ | |
| Rams: | $(\ )$ | |
| Jump set: | undefined | |
| Roots of unity: | $136 = (137 - 1)$ |
|
Intermediate fields
| $\Q_{137}(\sqrt{137\cdot 3})$, 137.1.3.2a1.1, 137.1.6.5a1.2, 137.1.9.8a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{137}$ |
|
| Relative Eisenstein polynomial: |
\( x^{18} + 411 \)
|
Ramification polygon
| Residual polynomials: | $z^{17} + 18 z^{16} + 16 z^{15} + 131 z^{14} + 46 z^{13} + 74 z^{12} + 69 z^{11} + 40 z^{10} + 55 z^9 + 122 z^8 + 55 z^7 + 40 z^6 + 69 z^5 + 74 z^4 + 46 z^3 + 131 z^2 + 16 z + 18$ |
| Associated inertia: | $6$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $108$ |
| Galois group: | $C_{18}:C_6$ (as 18T45) |
| Inertia group: | $C_{18}$ (as 18T1) |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $6$ |
| Galois tame degree: | $18$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.9444444444444444$ |
| Galois splitting model: | not computed |