Defining polynomial
\(x^{16} + 10\)
|
Invariants
Base field: | $\Q_{5}$ |
Degree $d$: | $16$ |
Ramification index $e$: | $16$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $15$ |
Discriminant root field: | $\Q_{5}(\sqrt{5\cdot 2})$ |
Root number: | $1$ |
$\Aut(K/\Q_{5})$: | $C_4$ |
This field is not Galois over $\Q_{5}.$ | |
Visible Artin slopes: | $[\ ]$ |
Visible Swan slopes: | $[\ ]$ |
Means: | $\langle\ \rangle$ |
Rams: | $(\ )$ |
Jump set: | undefined |
Roots of unity: | $4 = (5 - 1)$ |
Intermediate fields
$\Q_{5}(\sqrt{5\cdot 2})$, 5.1.4.3a1.2, 5.1.8.7a1.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
Unramified subfield: | $\Q_{5}$ |
Relative Eisenstein polynomial: |
\( x^{16} + 10 \)
|
Ramification polygon
Residual polynomials: | $z^{15} + z^{14} + 3 z^{10} + 3 z^9 + 3 z^5 + 3 z^4 + 1$ |
Associated inertia: | $4$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois degree: | $64$ |
Galois group: | $C_{16}:C_4$ (as 16T125) |
Inertia group: | $C_{16}$ (as 16T1) |
Wild inertia group: | $C_1$ |
Galois unramified degree: | $4$ |
Galois tame degree: | $16$ |
Galois Artin slopes: | $[\ ]$ |
Galois Swan slopes: | $[\ ]$ |
Galois mean slope: | $0.9375$ |
Galois splitting model: | not computed |