Defining polynomial
| \( x^{15} + 1395 x^{10} + 388244 x^{5} + 1759128759 \) |
Invariants
| Base field: | $\Q_{31}$ |
| Degree $d$ : | $15$ |
| Ramification exponent $e$ : | $5$ |
| Residue field degree $f$ : | $3$ |
| Discriminant exponent $c$ : | $12$ |
| Discriminant root field: | $\Q_{31}$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 31 })|$: | $15$ |
| This field is Galois and abelian over $\Q_{31}$. | |
Intermediate fields
| 31.3.0.1, 31.5.4.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 31.3.0.1 $\cong \Q_{31}(t)$ where $t$ is a root of \( x^{3} - x + 9 \) |
| Relative Eisenstein polynomial: | $ x^{5} - 31 t^{5} \in\Q_{31}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_{15}$ (as 15T1) |
| Inertia group: | Intransitive group isomorphic to $C_5$ |
| Unramified degree: | $3$ |
| Tame degree: | $5$ |
| Wild slopes: | None |
| Galois mean slope: | $4/5$ |
| Galois splitting model: | Not computed |