Defining polynomial
\(x^{15} - 45 x^{12} + 45 x^{11} + 15 x^{10} + 750 x^{9} + 1950 x^{8} + 1425 x^{7} + 45075 x^{6} + 15825 x^{5} + 43125 x^{4} + 7125 x^{3} + 8250 x^{2} + 1125 x + 125\) |
Invariants
Base field: | $\Q_{5}$ |
Degree $d$: | $15$ |
Ramification exponent $e$: | $5$ |
Residue field degree $f$: | $3$ |
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: | $[5/4]$ |
Intermediate fields
5.3.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 5.3.0.1 $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{3} + 3 x + 3 \) |
Relative Eisenstein polynomial: | \( x^{5} + \left(15 t^{2} + 5 t + 15\right) x^{2} + \left(20 t + 15\right) x + 5 \) $\ \in\Q_{5}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z + t + 2$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
Galois group: | $C_5^3:C_{12}$ (as 15T38) |
Inertia group: | Intransitive group isomorphic to $C_5^3:C_4$ |
Wild inertia group: | $C_5^3$ |
Unramified degree: | $3$ |
Tame degree: | $4$ |
Wild slopes: | $[5/4, 5/4, 5/4]$ |
Galois mean slope: | $623/500$ |
Galois splitting model: | $x^{15} + 10 x^{13} + 5 x^{11} - 18 x^{10} - 100 x^{9} - 190 x^{8} - 125 x^{7} - 630 x^{6} + 205 x^{5} - 850 x^{4} + 625 x^{3} - 750 x^{2} + 1080 x - 104$ |