Defining polynomial
\(x^{15} - 24 x^{14} + 414 x^{13} - 2469 x^{12} + 8325 x^{11} - 16038 x^{10} + 58167 x^{9} - 83187 x^{8} + 193671 x^{7} - 318789 x^{6} + 326916 x^{5} - 881847 x^{4} + 227853 x^{3} - 1376838 x^{2} - 1053891\) |
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: | \( x^{3} + \left(3 t^{4} + 3 t^{3} + 3 t^{2} + 3 t\right) x^{2} + 9 t^{3} + 9 t^{2} + 18 t + 3 \) $\ \in\Q_{3}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{2} + t^{4} + t^{3} + t^{2} + t$ |
Associated inertia: | $2$ |
Indices of inseparability: | $[2, 0]$ |
Invariants of the Galois closure
Galois group: | $C_3^4:C_{10}$ (as 15T33) |
Inertia group: | Intransitive group isomorphic to $C_3^4$ |
Wild inertia group: | $C_3^4$ |
Unramified degree: | $10$ |
Tame degree: | $1$ |
Wild slopes: | $[2, 2, 2, 2]$ |
Galois mean slope: | $160/81$ |
Galois splitting model: | $x^{15} - 12 x^{13} - 36 x^{12} - 81 x^{11} + 174 x^{10} + 1474 x^{9} + 2700 x^{8} - 114 x^{7} - 8430 x^{6} - 13743 x^{5} - 4476 x^{4} + 11296 x^{3} + 15768 x^{2} + 5568 x - 4192$ |