Defining polynomial
\(x^{15} + 6 x^{14} + 81 x^{13} + 1887 x^{12} + 12060 x^{11} + 34182 x^{10} + 145593 x^{9} + 446472 x^{8} + 538974 x^{7} + 2789910 x^{6} + 930123 x^{5} + 7395705 x^{4} + 1208601 x^{3} + 9607977 x^{2} + 7056234\) |
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: | \( x^{3} + \left(3 t^{4} + 3 t^{3} + 3 t^{2} + 6\right) x^{2} + 9 t^{3} + 21 \) $\ \in\Q_{3}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{2} + t^{4} + t^{3} + t^{2} + 2$ |
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: | $x^{15} + 3 x^{13} - 22 x^{12} - 333 x^{11} + 1122 x^{10} + 1756 x^{9} - 10296 x^{8} + 5523 x^{7} + 25894 x^{6} - 36783 x^{5} - 5742 x^{4} + 20944 x^{3} + 34056 x^{2} - 59664 x + 23584$ |