Defining polynomial
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$( x^{10} + x^{6} + x^{5} + x^{3} + x^{2} + x + 1 )^{2} + 4 x^{7} ( x^{10} + x^{6} + x^{5} + x^{3} + x^{2} + x + 1 ) + 8 x^{5} + 2$
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $20$ |
| Ramification index $e$: | $2$ |
| Residue field degree $f$: | $10$ |
| Discriminant exponent $c$: | $30$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-5})$ |
| Root number: | $i$ |
| $\Aut(K/\Q_{2})$: | $C_2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[3]$ |
| Visible Swan slopes: | $[2]$ |
| Means: | $\langle1\rangle$ |
| Rams: | $(2)$ |
| Jump set: | $[1, 3]$ |
| Roots of unity: | $2046 = (2^{ 10 } - 1) \cdot 2$ |
Intermediate fields
| $\Q_{2}(\sqrt{5})$, 2.5.1.0a1.1, 2.10.1.0a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 2.10.1.0a1.1 $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{10} + x^{6} + x^{5} + x^{3} + x^{2} + x + 1 \)
|
| Relative Eisenstein polynomial: |
\( x^{2} + \left(4 t^{7} + 4 t^{4} + 4 t^{3} + 4 t + 4\right) x + 8 t^{5} + 2 \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + (t^8 + t^4 + 1)$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[2, 0]$ |
Invariants of the Galois closure
| Galois degree: | $10240$ |
| Galois group: | $C_2^{10}.C_{10}$ (as 20T409) |
| Inertia group: | not computed |
| Wild inertia group: | not computed |
| Galois unramified degree: | not computed |
| Galois tame degree: | not computed |
| Galois Artin slopes: | not computed |
| Galois Swan slopes: | not computed |
| Galois mean slope: | not computed |
| Galois splitting model: | not computed |