Defining polynomial
\(x^{15} + 59\) |
Invariants
Base field: | $\Q_{59}$ |
Degree $d$: | $15$ |
Ramification exponent $e$: | $15$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $14$ |
Discriminant root field: | $\Q_{59}(\sqrt{2})$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 59 }) }$: | $1$ |
This field is not Galois over $\Q_{59}.$ | |
Visible slopes: | None |
Intermediate fields
59.3.2.1, 59.5.4.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{59}$ |
Relative Eisenstein polynomial: | \( x^{15} + 59 \) |
Ramification polygon
Residual polynomials: | $z^{14} + 15z^{13} + 46z^{12} + 42z^{11} + 8z^{10} + 53z^{9} + 49z^{8} + 4z^{7} + 4z^{6} + 49z^{5} + 53z^{4} + 8z^{3} + 42z^{2} + 46z + 15$ |
Associated inertia: | $2$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $D_{15}$ (as 15T2) |
Inertia group: | $C_{15}$ (as 15T1) |
Wild inertia group: | $C_1$ |
Unramified degree: | $2$ |
Tame degree: | $15$ |
Wild slopes: | None |
Galois mean slope: | $14/15$ |
Galois splitting model: | Not computed |