Defining polynomial
\(x^{15} + 3 x^{14} + 15 x^{13} + 24 x^{12} + 24 x^{11} + 18 x^{10} + 10 x^{9} + 21 x^{8} + 9 x^{7} + 15 x^{6} + 9 x^{5} + 3 x^{4} + 13 x^{3} + 18 x + 7\)  |
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $15$ |
| Ramification exponent $e$: | $3$ |
| Residue field degree $f$: | $5$ |
| Discriminant exponent $c$: | $15$ |
| Discriminant root field: | $\Q_{3}(\sqrt{3})$ |
| Root number: | $-i$ |
| $|\Aut(K/\Q_{ 3 })|$: | $1$ |
| This field is not Galois over $\Q_{3}.$ | |
Intermediate fields
| 3.5.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 3.5.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{5} - x + 1 \) |
| Relative Eisenstein polynomial: | \( x^{3} + \left(6 t^{2} + 3 t + 3\right) x^{2} + \left(6 t^{4} + 3 t^{2}\right) x + 3 t^{3} + 3 t^{2} + 6 \)$\ \in\Q_{3}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | 15T44 |
| Inertia group: | Intransitive group isomorphic to $C_3^2:(C_3^3:C_2)$ |
| Unramified degree: | $5$ |
| Tame degree: | $2$ |
| Wild slopes: | [3/2, 3/2, 3/2, 3/2, 3/2] |
| Galois mean slope: | $727/486$ |
| Galois splitting model: | $x^{15} - 3 x^{14} - 276 x^{13} + 10178 x^{12} - 354816 x^{11} + 1537788 x^{10} + 90568692 x^{9} - 2306546520 x^{8} + 39691030482 x^{7} - 140030871528 x^{6} - 4956080879892 x^{5} + 23119953452244 x^{4} - 65396756162112 x^{3} + 12688679327651868 x^{2} - 155866955794490664 x + 514602372876454411$ |