Defining polynomial
|
\(x^{16} + 8 x^{13} + 8 x^{11} + 4 x^{10} + 8 x^{9} + 2 x^{8} + 8 x^{5} + 6\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $16$ |
| Ramification index $e$: | $16$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $52$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{2})$: | $C_2^2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[2, 3, 3, 4]$ |
| Visible Swan slopes: | $[1,2,2,3]$ |
| Means: | $\langle\frac{1}{2}, \frac{5}{4}, \frac{13}{8}, \frac{37}{16}\rangle$ |
| Rams: | $(1, 3, 3, 11)$ |
| Jump set: | $[1, 2, 4, 8, 32]$ |
| Roots of unity: | $2$ |
Intermediate fields
| $\Q_{2}(\sqrt{-5})$, 2.1.4.8b1.4, 2.1.8.20d1.7, 2.1.8.24c1.34, 2.1.8.24c1.38 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{2}$ |
| Relative Eisenstein polynomial: |
\( x^{16} + 8 x^{13} + 8 x^{11} + 4 x^{10} + 8 x^{9} + 2 x^{8} + 8 x^{5} + 6 \)
|
Ramification polygon
| Residual polynomials: | $z^8 + 1$,$z^6 + 1$,$z + 1$ |
| Associated inertia: | $1$,$2$,$1$ |
| Indices of inseparability: | $[37, 26, 24, 8, 0]$ |
Invariants of the Galois closure
| Galois degree: | $128$ |
| Galois group: | $C_2^4.D_4$ (as 16T254) |
| Inertia group: | $C_4^2:C_2$ (as 16T17) |
| Wild inertia group: | $C_4^2:C_2$ |
| Galois unramified degree: | $4$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[2, 2, 3, 3, 4]$ |
| Galois Swan slopes: | $[1,1,2,2,3]$ |
| Galois mean slope: | $3.3125$ |
| Galois splitting model: |
$x^{16} - 8 x^{15} + 32 x^{14} - 64 x^{13} + 56 x^{12} + 352 x^{11} - 1440 x^{10} + 3168 x^{9} - 3696 x^{8} - 1056 x^{7} + 15200 x^{6} - 36736 x^{5} + 52096 x^{4} - 50432 x^{3} + 35968 x^{2} - 17536 x + 4176$
|