Defining polynomial
|
\(x^{18} + 3 x^{12} + 9 x^{5} + 9 x^{2} + 15\)
|
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $18$ |
| Ramification index $e$: | $18$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $37$ |
| Discriminant root field: | $\Q_{3}(\sqrt{3})$ |
| Root number: | $-i$ |
| $\Aut(K/\Q_{3})$: | $C_3$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible Artin slopes: | $[2, \frac{7}{3}]$ |
| Visible Swan slopes: | $[1,\frac{4}{3}]$ |
| Means: | $\langle\frac{2}{3}, \frac{10}{9}\rangle$ |
| Rams: | $(2, 4)$ |
| Jump set: | undefined |
| Roots of unity: | $2 = (3 - 1)$ |
Intermediate fields
| $\Q_{3}(\sqrt{3})$, 3.1.3.4a2.2, 3.1.6.9a1.8 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{3}$ |
| Relative Eisenstein polynomial: |
\( x^{18} + 3 x^{12} + 9 x^{5} + 9 x^{2} + 15 \)
|
Ramification polygon
| Residual polynomials: | $z^9 + 2$,$2 z^6 + 1$,$z^2 + 2$ |
| Associated inertia: | $1$,$1$,$1$ |
| Indices of inseparability: | $[20, 12, 0]$ |
Invariants of the Galois closure
| Galois degree: | $486$ |
| Galois group: | $C_3^4:C_6$ (as 18T163) |
| Inertia group: | not computed |
| Wild inertia group: | not computed |
| Galois unramified degree: | not computed |
| Galois tame degree: | not computed |
| Galois Artin slopes: | not computed |
| Galois Swan slopes: | not computed |
| Galois mean slope: | not computed |
| Galois splitting model: | not computed |