Defining polynomial
\(x^{15} + 6 x^{14} + 189 x^{13} + 321 x^{12} - 1980 x^{11} + 9072 x^{10} - 55935 x^{9} - 77193 x^{8} + 33291 x^{7} + 582984 x^{6} + 438777 x^{5} + 3456918 x^{4} + 1635066 x^{3} + 3636738 x^{2} - 8018757\) |
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: | \( x^{3} + \left(3 t^{4} + 6 t^{2} + 3 t + 6\right) x^{2} + 9 t^{4} + 9 t^{3} + 18 t + 3 \) $\ \in\Q_{3}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{2} + t^{4} + 2t^{2} + t + 2$ |
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: | $x^{15} - 6 x^{13} - 28 x^{12} - 144 x^{11} - 156 x^{10} - 167 x^{9} - 36 x^{8} + 1806 x^{7} - 14 x^{6} + 801 x^{5} + 1224 x^{4} - 13460 x^{3} + 10944 x^{2} + 1248 x - 2144$ |