Defining polynomial
\(x^{15} + 15 x^{2} + 5 x + 5\) |
Invariants
Base field: | $\Q_{5}$ |
Degree $d$: | $15$ |
Ramification exponent $e$: | $15$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $15$ |
Discriminant root field: | $\Q_{5}(\sqrt{5})$ |
Root number: | $-1$ |
$\card{ \Aut(K/\Q_{ 5 }) }$: | $1$ |
This field is not Galois over $\Q_{5}.$ | |
Visible slopes: | $[13/12]$ |
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: | \( x^{15} + 15 x^{2} + 5 x + 5 \) |
Ramification polygon
Residual polynomials: | $3z + 4$,$z^{10} + 3z^{5} + 3$ |
Associated inertia: | $1$,$2$ |
Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
Galois group: | $C_5^2:(C_4\times S_3)$ (as 15T27) |
Inertia group: | $C_5^2:C_{12}$ (as 15T19) |
Wild inertia group: | $C_5^2$ |
Unramified degree: | $2$ |
Tame degree: | $12$ |
Wild slopes: | $[13/12, 13/12]$ |
Galois mean slope: | $323/300$ |
Galois splitting model: | $x^{15} + 105 x^{13} - 40 x^{12} - 8595 x^{11} - 338532 x^{10} + 1278915 x^{9} + 55614720 x^{8} + 792363915 x^{7} + 5662612560 x^{6} + 31423672683 x^{5} + 135967366260 x^{4} + 257447443320 x^{3} + 417362109120 x^{2} + 1242723555180 x - 4920495387984$ |