Defining polynomial
\(x^{15} - 9 x^{14} + 369 x^{13} + 150 x^{12} - 486 x^{11} - 2430 x^{10} - 8982 x^{9} + 39366 x^{8} + 62208 x^{7} + 497934 x^{6} + 288441 x^{5} + 1624698 x^{4} + 376569 x^{3} + 2388204 x^{2} + 1235898\) |
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: | \( x^{3} + \left(3 t^{4} + 3 t^{3} + 6 t^{2} + 3 t + 3\right) x^{2} + 18 t^{2} + 9 t + 3 \) $\ \in\Q_{3}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{2} + t^{4} + t^{3} + 2t^{2} + t + 1$ |
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: | $x^{15} - 12 x^{13} - 16 x^{12} - 81 x^{11} - 282 x^{10} - 141 x^{9} + 342 x^{8} + 1449 x^{7} + 4560 x^{6} + 5553 x^{5} + 3378 x^{4} + 6432 x^{3} + 9288 x^{2} + 3024 x - 1376$ |