Defining polynomial
\(x^{15} + 13454 x^{9} + 45252529 x^{3} + 22445254384\) |
Invariants
Base field: | $\Q_{31}$ |
Degree $d$: | $15$ |
Ramification exponent $e$: | $3$ |
Residue field degree $f$: | $5$ |
Discriminant exponent $c$: | $10$ |
Discriminant root field: | $\Q_{31}$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 31 }) }$: | $15$ |
This field is Galois and abelian over $\Q_{31}.$ | |
Visible slopes: | None |
Intermediate fields
31.3.2.2, 31.5.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 31.5.0.1 $\cong \Q_{31}(t)$ where $t$ is a root of \( x^{5} + 7 x + 28 \) |
Relative Eisenstein polynomial: | \( x^{3} + 31 t^{2} \) $\ \in\Q_{31}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{2} + 3z + 3$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $C_{15}$ (as 15T1) |
Inertia group: | Intransitive group isomorphic to $C_3$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $5$ |
Tame degree: | $3$ |
Wild slopes: | None |
Galois mean slope: | $2/3$ |
Galois splitting model: | Not computed |