Defining polynomial
| \( x^{15} - 3 x^{14} + 3 x^{13} + 3 x^{11} + 3 x^{10} - 3 x^{7} - 3 x^{6} - 3 x^{5} + 3 x^{3} - 3 x^{2} - 3 x - 3 \) |
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$ : | $15$ |
| Ramification exponent $e$ : | $15$ |
| Residue field degree $f$ : | $1$ |
| 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
| 3.5.4.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{3}$ |
| Relative Eisenstein polynomial: | \( x^{15} - 3 x^{14} + 3 x^{13} + 3 x^{11} + 3 x^{10} - 3 x^{7} - 3 x^{6} - 3 x^{5} + 3 x^{3} - 3 x^{2} - 3 x - 3 \) |
Invariants of the Galois closure
| Galois group: | 15T52 |
| Inertia group: | 15T33 |
| Unramified degree: | $4$ |
| Tame degree: | $10$ |
| Wild slopes: | [11/10, 11/10, 11/10, 11/10] |
| Galois mean slope: | $889/810$ |
| Galois splitting model: | $x^{15} - 12 x^{14} - 24 x^{13} + 685 x^{12} - 840 x^{11} - 11532 x^{10} + 42694 x^{9} + 27738 x^{8} - 582210 x^{7} + 1878920 x^{6} - 3095874 x^{5} + 2807568 x^{4} - 1382389 x^{3} + 614850 x^{2} - 552630 x + 256955$ |