Defining polynomial
|
\(x^{15} + 36 x^{13} - 57 x^{12} + 1026 x^{11} - 729 x^{10} + 11619 x^{9} + 13851 x^{8} + 85050 x^{7} + 227475 x^{6} + 224532 x^{5} + 671652 x^{4} + 968031 x^{3} + 1350837 x^{2} + 2809080\)
|
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: |
\( x^{3} + 3 t^{2} x^{2} + 9 t^{4} + 18 t^{3} + 18 t^{2} + 3 \)
$\ \in\Q_{3}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^{2} + t^{2}$ |
| 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: |
$x^{15} - 3 x^{14} - 45 x^{13} + 308 x^{12} - 138 x^{11} - 5727 x^{10} + 26545 x^{9} - 36816 x^{8} - 118632 x^{7} + 735536 x^{6} - 1923423 x^{5} + 3135012 x^{4} - 3405457 x^{3} + 2441040 x^{2} - 1054827 x + 206933$
|