Defining polynomial
\(x^{16} + 8 x^{15} + 4 x^{14} + 4 x^{12} + 8 x^{11} + 2 x^{8} + 8 x^{6} + 4 x^{4} + 8 x^{2} + 10\)
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Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $16$ |
Ramification index $e$: | $16$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $58$ |
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: | $[2, 3, \frac{7}{2}, \frac{9}{2}]$ |
Visible Swan slopes: | $[1,2,\frac{5}{2},\frac{7}{2}]$ |
Means: | $\langle\frac{1}{2}, \frac{5}{4}, \frac{15}{8}, \frac{43}{16}\rangle$ |
Rams: | $(1, 3, 5, 13)$ |
Jump set: | $[1, 7, 14, 32, 48]$ |
Roots of unity: | $4 = 2^{ 2 }$ |
Intermediate fields
$\Q_{2}(\sqrt{-1})$, $\Q_{2}(\sqrt{2\cdot 5})$, $\Q_{2}(\sqrt{-2\cdot 5})$, 2.1.4.8b1.2, 2.1.4.9a1.2, 2.1.4.9a1.1, 2.1.8.22d1.16, 2.1.8.27a1.31, 2.1.8.27a1.32 |
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: |
\( x^{16} + 8 x^{15} + 4 x^{14} + 4 x^{12} + 8 x^{11} + 2 x^{8} + 8 x^{6} + 4 x^{4} + 8 x^{2} + 10 \)
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Ramification polygon
Residual polynomials: | $z^8 + 1$,$z^4 + 1$,$z^2 + 1$,$z + 1$ |
Associated inertia: | $1$,$1$,$1$,$1$ |
Indices of inseparability: | $[43, 30, 20, 8, 0]$ |
Invariants of the Galois closure
Galois degree: | $32$ |
Galois group: | $C_2\times \SD_{16}$ (as 16T48) |
Inertia group: | $\SD_{16}$ (as 16T12) |
Wild inertia group: | $\SD_{16}$ |
Galois unramified degree: | $2$ |
Galois tame degree: | $1$ |
Galois Artin slopes: | $[2, 3, \frac{7}{2}, \frac{9}{2}]$ |
Galois Swan slopes: | $[1,2,\frac{5}{2},\frac{7}{2}]$ |
Galois mean slope: | $3.625$ |
Galois splitting model: |
$x^{16} - 8 x^{15} - 96 x^{14} + 616 x^{13} + 3564 x^{12} - 17240 x^{11} - 64352 x^{10} + 219320 x^{9} + 558694 x^{8} - 1319320 x^{7} - 1923040 x^{6} + 3812280 x^{5} + 1717324 x^{4} - 3527752 x^{3} - 213984 x^{2} + 901224 x - 121743$
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