Defining polynomial
\(x^{15} + 78 x^{14} + 39 x^{13} + 49 x^{12} + 24 x^{11} + 36 x^{10} + 2 x^{9} + 54 x^{8} + 69 x^{7} + 47 x^{6} + 18 x^{5} + 15 x^{4} + 36 x^{3} + 36 x^{2} + 63 x + 73\)  |
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $15$ |
| Ramification exponent $e$: | $3$ |
| Residue field degree $f$: | $5$ |
| Discriminant exponent $c$: | $20$ |
| Discriminant root field: | $\Q_{3}$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 3 })|$: | $15$ |
| This field is Galois and abelian over $\Q_{3}.$ | |
Intermediate fields
| 3.3.4.2, 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: | \( x^{3} + \left(15 t^{4} + 24 t^{3} + 15 t^{2} + 9\right) x^{2} + \left(18 t^{4} + 18 t^{2} + 9 t + 18\right) x + 12 t^{3} + 6 t^{2} + 6 \)$\ \in\Q_{3}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_{15}$ (as 15T1) |
| Inertia group: | Intransitive group isomorphic to $C_3$ |
| Unramified degree: | $5$ |
| Tame degree: | $1$ |
| Wild slopes: | [2] |
| Galois mean slope: | $4/3$ |
| Galois splitting model: | $x^{15} - 3 x^{14} - 36 x^{13} + 151 x^{12} + 306 x^{11} - 2250 x^{10} + 1258 x^{9} + 10467 x^{8} - 18066 x^{7} - 7856 x^{6} + 39705 x^{5} - 23016 x^{4} - 8527 x^{3} + 5730 x^{2} + 2244 x + 199$ |