Defining polynomial
\(x^{15} + 6 x^{14} + 6 x^{13} + 3 x^{12} + 6 x^{11} + 6 x^{10} + 6 x^{9} + 3 x^{6} + 6 x^{3} + 12\) |
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: | \( x^{15} + 6 x^{14} + 6 x^{13} + 3 x^{12} + 6 x^{11} + 6 x^{10} + 6 x^{9} + 3 x^{6} + 6 x^{3} + 12 \) |
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: | $x^{15} - 105 x^{13} - 70 x^{12} + 4410 x^{11} + 7350 x^{10} - 90650 x^{9} - 246960 x^{8} + 833490 x^{7} + 3615220 x^{6} - 1663893 x^{5} - 26615085 x^{4} - 45655015 x^{3} - 21176820 x^{2} + 13109460 x + 11563216$ |