Defining polynomial
\(x^{15} + 60 x^{14} + 1200 x^{13} + 8000 x^{12} + 15 x^{10} + 600 x^{9} + 6000 x^{8} + 7575 x^{5} + 151500 x^{4} - 337375\) |
Invariants
Base field: | $\Q_{5}$ |
Degree $d$: | $15$ |
Ramification exponent $e$: | $5$ |
Residue field degree $f$: | $3$ |
Discriminant exponent $c$: | $24$ |
Discriminant root field: | $\Q_{5}$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 5 }) }$: | $15$ |
This field is Galois and abelian over $\Q_{5}.$ | |
Visible slopes: | $[2]$ |
Intermediate fields
5.3.0.1, 5.5.8.2 |
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: | \( x^{5} + 20 x^{4} + 50 t + 5 \) $\ \in\Q_{5}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{4} + 4$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[4, 0]$ |