Defining polynomial
\(x^{15} - 33 x^{14} + 741 x^{13} - 3054 x^{12} - 22212 x^{11} + 52065 x^{10} + 398718 x^{9} + 775089 x^{8} + 832599 x^{7} + 626886 x^{6} + 184761 x^{5} - 104409 x^{4} - 34101 x^{3} + 16281 x^{2} - 2673 x + 243\) |
Invariants
Base field: | $\Q_{3}$ |
Degree $d$: | $15$ |
Ramification exponent $e$: | $3$ |
Residue field degree $f$: | $5$ |
Discriminant exponent $c$: | $15$ |
Discriminant root field: | $\Q_{3}(\sqrt{3\cdot 2})$ |
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: | \( x^{3} + \left(6 t^{4} + 3 t^{3} + 6 t^{2} + 3 t + 3\right) x^{2} + \left(6 t^{4} + 6 t^{2} + 6 t + 3\right) x + 3 \) $\ \in\Q_{3}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z + t^{4} + t^{2} + t + 2$ |
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: | $x^{15} - 10 x^{12} - 125 x^{9} + 1625 x^{6} - 4375 x^{3} + 3125$ |