Defining polynomial
\(x^{15} + 146\) |
Invariants
Base field: | $\Q_{73}$ |
Degree $d$: | $15$ |
Ramification exponent $e$: | $15$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $14$ |
Discriminant root field: | $\Q_{73}(\sqrt{5})$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 73 }) }$: | $3$ |
This field is not Galois over $\Q_{73}.$ | |
Visible slopes: | None |
Intermediate fields
73.3.2.3, 73.5.4.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{73}$ |
Relative Eisenstein polynomial: | \( x^{15} + 146 \) |
Ramification polygon
Residual polynomials: | $z^{14} + 15z^{13} + 32z^{12} + 17z^{11} + 51z^{10} + 10z^{9} + 41z^{8} + 11z^{7} + 11z^{6} + 41z^{5} + 10z^{4} + 51z^{3} + 17z^{2} + 32z + 15$ |
Associated inertia: | $4$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $C_3\times F_5$ (as 15T8) |
Inertia group: | $C_{15}$ (as 15T1) |
Wild inertia group: | $C_1$ |
Unramified degree: | $4$ |
Tame degree: | $15$ |
Wild slopes: | None |
Galois mean slope: | $14/15$ |
Galois splitting model: | Not computed |