Defining polynomial
\(x^{9} + 6 x^{7} + 123 x^{6} + 12 x^{5} + 78 x^{4} - 6127 x^{3} + 492 x^{2} - 6888 x + 69105\) |
Invariants
Base field: | $\Q_{23}$ |
Degree $d$: | $9$ |
Ramification exponent $e$: | $3$ |
Residue field degree $f$: | $3$ |
Discriminant exponent $c$: | $6$ |
Discriminant root field: | $\Q_{23}(\sqrt{5})$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 23 }) }$: | $3$ |
This field is not Galois over $\Q_{23}.$ | |
Visible slopes: | None |
Intermediate fields
23.3.2.1, 23.3.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 23.3.0.1 $\cong \Q_{23}(t)$ where $t$ is a root of \( x^{3} + 2 x + 18 \) |
Relative Eisenstein polynomial: | \( x^{3} + 23 \) $\ \in\Q_{23}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{2} + 3z + 3$ |
Associated inertia: | $2$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $C_3\times S_3$ (as 9T4) |
Inertia group: | Intransitive group isomorphic to $C_3$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $6$ |
Tame degree: | $3$ |
Wild slopes: | None |
Galois mean slope: | $2/3$ |
Galois splitting model: | Not computed |