Defining polynomial
|
\(x^{15} + 30 x^{14} + 360 x^{13} + 2220 x^{12} + 7920 x^{11} + 20736 x^{10} + 54576 x^{9} + 127008 x^{8} + 295488 x^{7} + 682614 x^{6} + 1099656 x^{5} + 3298968 x^{4} + 2128842 x^{3} + 12773052 x^{2} + 9410175\)
|
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $15$ |
| Ramification exponent $e$: | $3$ |
| Residue field degree $f$: | $5$ |
| Discriminant exponent $c$: | $20$ |
| Discriminant root field: | $\Q_{3}$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 3 }) }$: | $15$ |
| This field is Galois and abelian over $\Q_{3}.$ | |
| Visible slopes: | $[2]$ |
Intermediate fields
| 3.3.4.3, 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: |
\( x^{3} + 6 x^{2} + 18 t^{3} + 9 t + 12 \)
$\ \in\Q_{3}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^{2} + 2$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[2, 0]$ |