Defining polynomial
\(x^{15} - 9 x^{14} + 720 x^{13} + 1185 x^{12} + 12312 x^{11} + 2673 x^{10} + 119025 x^{9} + 76059 x^{8} + 544077 x^{7} + 269271 x^{6} + 897156 x^{5} - 200718 x^{4} + 583119 x^{3} - 1163484 x^{2} - 1000674\) |
Invariants
Base field: | $\Q_{3}$ |
Degree $d$: | $15$ |
Ramification exponent $e$: | $3$ |
Residue field degree $f$: | $5$ |
Discriminant exponent $c$: | $20$ |
Discriminant root field: | $\Q_{3}(\sqrt{2})$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 3 }) }$: | $1$ |
This field is not Galois over $\Q_{3}.$ | |
Visible slopes: | $[2]$ |
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 \( x^{5} + 2 x + 1 \) |
Relative Eisenstein polynomial: | \( x^{3} + \left(3 t^{4} + 6 t^{3} + 6 t^{2} + 6 t + 3\right) x^{2} + 9 t^{4} + 9 t^{3} + 9 t + 3 \) $\ \in\Q_{3}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{2} + t^{4} + 2t^{3} + 2t^{2} + 2t + 1$ |
Associated inertia: | $2$ |
Indices of inseparability: | $[2, 0]$ |
Invariants of the Galois closure
Galois group: | $C_7^3:C_6$ (as 15T44) |
Inertia group: | Intransitive group isomorphic to $C_3^5$ |
Wild inertia group: | $C_3^5$ |
Unramified degree: | $10$ |
Tame degree: | $1$ |
Wild slopes: | $[2, 2, 2, 2, 2]$ |
Galois mean slope: | $484/243$ |
Galois splitting model: | $x^{15} - 12 x^{13} - 14 x^{12} + 18 x^{11} + 42 x^{10} + 99 x^{9} + 522 x^{8} + 1338 x^{7} - 246 x^{6} - 6615 x^{5} - 7446 x^{4} + 5752 x^{3} + 11016 x^{2} - 2352 x - 6304$ |