Defining polynomial
\(x^{18} - 178 x^{15} + 1275281 x^{12} + 4623186702 x^{9} + 247580882986 x^{6} - 1423935159495 x^{3} + 4472831618649\) |
Invariants
Base field: | $\Q_{89}$ |
Degree $d$: | $18$ |
Ramification exponent $e$: | $3$ |
Residue field degree $f$: | $6$ |
Discriminant exponent $c$: | $12$ |
Discriminant root field: | $\Q_{89}(\sqrt{3})$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 89 }) }$: | $9$ |
This field is not Galois over $\Q_{89}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{89}(\sqrt{3})$, 89.3.0.1, 89.6.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 89.6.0.1 $\cong \Q_{89}(t)$ where $t$ is a root of \( x^{6} + x^{4} + 82 x^{3} + 80 x^{2} + 15 x + 3 \) |
Relative Eisenstein polynomial: | \( x^{3} + 89 t^{2} \) $\ \in\Q_{89}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{2} + 3z + 3$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $S_3\times C_9$ (as 18T16) |
Inertia group: | Intransitive group isomorphic to $C_3$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $18$ |
Tame degree: | $3$ |
Wild slopes: | None |
Galois mean slope: | $2/3$ |
Galois splitting model: | Not computed |