Defining polynomial
\(x^{15} + 15 x^{14} + 20 x^{12} + 5 x^{10} + 20 x^{5} + 105\) |
Invariants
Base field: | $\Q_{5}$ |
Degree $d$: | $15$ |
Ramification exponent $e$: | $15$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $26$ |
Discriminant root field: | $\Q_{5}(\sqrt{2})$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 5 }) }$: | $5$ |
This field is not Galois over $\Q_{5}.$ | |
Visible slopes: | $[2]$ |
Intermediate fields
5.3.2.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{5}$ |
Relative Eisenstein polynomial: | \( x^{15} + 15 x^{14} + 20 x^{12} + 5 x^{10} + 20 x^{5} + 105 \) |
Ramification polygon
Residual polynomials: | $3z^{4} + 2$,$z^{10} + 3z^{5} + 3$ |
Associated inertia: | $1$,$2$ |
Indices of inseparability: | $[12, 0]$ |
Invariants of the Galois closure
Galois group: | $C_5\wr S_3$ (as 15T32) |
Inertia group: | $C_5\wr C_3$ (as 15T25) |
Wild inertia group: | $C_5^3$ |
Unramified degree: | $2$ |
Tame degree: | $3$ |
Wild slopes: | $[5/3, 5/3, 2]$ |
Galois mean slope: | $722/375$ |
Galois splitting model: | $x^{15} - 165 x^{13} - 660 x^{12} + 10835 x^{11} + 100793 x^{10} - 245025 x^{9} - 5932630 x^{8} - 9222620 x^{7} + 166380445 x^{6} + 697559313 x^{5} - 2167733150 x^{4} - 14720746865 x^{3} + 9336818590 x^{2} + 84853592560 x + 65947005091$ |