Defining polynomial
\(x^{9} - 152 x^{6} + 5776 x^{3} + 1982251\) |
Invariants
Base field: | $\Q_{19}$ |
Degree $d$: | $9$ |
Ramification exponent $e$: | $3$ |
Residue field degree $f$: | $3$ |
Discriminant exponent $c$: | $6$ |
Discriminant root field: | $\Q_{19}$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 19 }) }$: | $9$ |
This field is Galois and abelian over $\Q_{19}.$ | |
Visible slopes: | None |
Intermediate fields
19.3.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 19.3.0.1 $\cong \Q_{19}(t)$ where $t$ is a root of \( x^{3} + 4 x + 17 \) |
Relative Eisenstein polynomial: | \( x^{3} + 19 t^{2} \) $\ \in\Q_{19}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{2} + 3z + 3$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $C_9$ (as 9T1) |
Inertia group: | Intransitive group isomorphic to $C_3$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $3$ |
Tame degree: | $3$ |
Wild slopes: | None |
Galois mean slope: | $2/3$ |
Galois splitting model: | Not computed |