Defining polynomial
\(x^{15} + 6 x^{14} + 3 x^{13} + 6 x^{12} + 3 x^{11} + 6 x^{10} + 3 x^{9} + 3 x^{6} + 6 x^{3} + 21\) |
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: | \( x^{15} + 6 x^{14} + 3 x^{13} + 6 x^{12} + 3 x^{11} + 6 x^{10} + 3 x^{9} + 3 x^{6} + 6 x^{3} + 21 \) |
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: | $[7/5, 7/5, 7/5, 7/5, 2]$ |
Galois mean slope: | $728/405$ |
Galois splitting model: | $x^{15} - 75 x^{13} - 325 x^{12} + 1965 x^{11} + 27279 x^{10} - 229700 x^{9} + 543690 x^{8} + 384750 x^{7} - 2831915 x^{6} - 881460 x^{5} + 10555260 x^{4} + 2332700 x^{3} - 24905175 x^{2} - 3856275 x + 28684948$ |