Defining polynomial
\(x^{15} + 25 x^{12} + 12 x^{11} + 9 x^{10} + 250 x^{9} - 900 x^{8} - 1302 x^{7} + 1322 x^{6} - 1773 x^{5} + 11325 x^{4} + 4989 x^{3} + 16494 x^{2} - 717 x + 2572\) |
Invariants
Base field: | $\Q_{5}$ |
Degree $d$: | $15$ |
Ramification exponent $e$: | $3$ |
Residue field degree $f$: | $5$ |
Discriminant exponent $c$: | $10$ |
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: | None |
Intermediate fields
5.3.2.1, 5.5.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 5.5.0.1 $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{5} + 4 x + 3 \) |
Relative Eisenstein polynomial: | \( x^{3} + 5 \) $\ \in\Q_{5}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{2} + 3z + 3$ |
Associated inertia: | $2$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $C_5\times S_3$ (as 15T4) |
Inertia group: | Intransitive group isomorphic to $C_3$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $10$ |
Tame degree: | $3$ |
Wild slopes: | None |
Galois mean slope: | $2/3$ |
Galois splitting model: | $x^{15} - 2 x^{14} - 3 x^{13} - 6 x^{12} + 46 x^{11} - 96 x^{10} + 66 x^{9} - 271 x^{8} + 802 x^{7} - 1192 x^{6} + 2483 x^{5} - 2152 x^{4} + 5024 x^{3} - 605 x^{2} - 3875 x - 131$ |