Defining polynomial
\(x^{15} + 6 x^{14} + 6 x^{13} + 3 x^{12} + 6 x^{11} + 6 x^{10} + 3\) |
Invariants
Base field: | $\Q_{3}$ |
Degree $d$: | $15$ |
Ramification exponent $e$: | $15$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $24$ |
Discriminant root field: | $\Q_{3}(\sqrt{2})$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 3 }) }$: | $3$ |
This field is not Galois over $\Q_{3}.$ | |
Visible slopes: | $[2]$ |
Intermediate fields
3.5.4.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{3}$ |
Relative Eisenstein polynomial: | \( x^{15} + 6 x^{14} + 6 x^{13} + 3 x^{12} + 6 x^{11} + 6 x^{10} + 3 \) |
Ramification polygon
Residual polynomials: | $2z^{2} + 1$,$z^{12} + 2z^{9} + z^{6} + z^{3} + 2$ |
Associated inertia: | $1$,$4$ |
Indices of inseparability: | $[10, 0]$ |
Invariants of the Galois closure
Galois group: | $C_3\wr F_5$ (as 15T56) |
Inertia group: | $C_3\wr C_5$ (as 15T36) |
Wild inertia group: | $C_3^5$ |
Unramified degree: | $4$ |
Tame degree: | $5$ |
Wild slopes: | $[9/5, 9/5, 9/5, 9/5, 2]$ |
Galois mean slope: | $2344/1215$ |
Galois splitting model: | $x^{15} - 105 x^{13} - 25 x^{12} + 4410 x^{11} + 2100 x^{10} - 93170 x^{9} - 66150 x^{8} + 1007685 x^{7} + 929530 x^{6} - 4824981 x^{5} - 5006085 x^{4} + 5455415 x^{3} + 1512630 x^{2} - 5654355 x + 2833180$ |