Defining polynomial
|
\(x^{15} - 24 x^{14} + 597 x^{13} - 4503 x^{12} + 18945 x^{11} - 48726 x^{10} + 86652 x^{9} - 73467 x^{8} - 27135 x^{7} + 114804 x^{6} + 44550 x^{5} - 8667 x^{4} + 4293 x^{3} + 7776 x^{2} + 2430 x + 243\)
|
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: | $-i$ |
| $\card{ \Aut(K/\Q_{ 3 }) }$: | $5$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible slopes: | $[3/2]$ |
Intermediate fields
| 3.3.3.1, 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: |
\( x^{3} + \left(3 t^{4} + 3 t^{3} + 6 t^{2} + 3 t\right) x^{2} + 6 x + 3 \)
$\ \in\Q_{3}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z + 1$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_5\times S_3$ (as 15T4) |
| Inertia group: | Intransitive group isomorphic to $S_3$ |
| Wild inertia group: | $C_3$ |
| Unramified degree: | $5$ |
| Tame degree: | $2$ |
| Wild slopes: | $[3/2]$ |
| Galois mean slope: | $7/6$ |
| Galois splitting model: |
$x^{15} - 3 x^{14} + 21 x^{13} - 31 x^{12} + 207 x^{11} - 102 x^{10} + 947 x^{9} + 1488 x^{8} + 75 x^{7} + 14565 x^{6} - 16692 x^{5} + 31335 x^{4} - 35754 x^{3} - 125298 x^{2} + 86121 x + 21473$
|