Defining polynomial
\(x^{15} - 18 x^{14} + 636 x^{13} - 1623 x^{12} - 378 x^{11} + 83475 x^{10} + 197730 x^{9} + 142074 x^{8} + 49653 x^{7} + 194913 x^{6} + 392931 x^{5} + 317520 x^{4} + 107811 x^{3} + 12393 x^{2} + 486 x + 243\) |
Invariants
Base field: | $\Q_{3}$ |
Degree $d$: | $15$ |
Ramification exponent $e$: | $3$ |
Residue field degree $f$: | $5$ |
Discriminant exponent $c$: | $15$ |
Discriminant root field: | $\Q_{3}(\sqrt{3})$ |
Root number: | $-i$ |
$\card{ \Aut(K/\Q_{ 3 }) }$: | $1$ |
This field is not Galois over $\Q_{3}.$ | |
Visible slopes: | $[3/2]$ |
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: | \( x^{3} + \left(6 t^{4} + 3 t^{3} + 6 t^{2} + 6 t + 6\right) x^{2} + \left(3 t^{4} + 6 t^{3} + 6 t^{2} + 3 t + 6\right) x + 3 \) $\ \in\Q_{3}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z + 2t^{4} + t^{3} + t^{2} + 2t + 1$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
Galois group: | $C_7^3:C_6$ (as 15T44) |
Inertia group: | Intransitive group isomorphic to $C_3^4:S_3$ |
Wild inertia group: | $C_3^5$ |
Unramified degree: | $5$ |
Tame degree: | $2$ |
Wild slopes: | $[3/2, 3/2, 3/2, 3/2, 3/2]$ |
Galois mean slope: | $727/486$ |
Galois splitting model: | $x^{15} - 3 x^{14} - 276 x^{13} + 8843 x^{12} - 295809 x^{11} + 1132749 x^{10} + 57567759 x^{9} - 1138085901 x^{8} + 15861753339 x^{7} + 49909480523 x^{6} - 359972447472 x^{5} - 90910666555560 x^{4} + 865603118772207 x^{3} + 8210364682792638 x^{2} - 125025136421986278 x + 359017702103139089$ |