Defining polynomial
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$( x^{3} + x + 1 )^{6} + 4 ( x^{3} + x + 1 )^{5} + 4 x ( x^{3} + x + 1 )^{3} + 10$
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Invariants
| Base field: | $\Q_{2}$ |
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| Degree $d$: | $18$ |
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| Ramification index $e$: | $6$ |
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| Residue field degree $f$: | $3$ |
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| Discriminant exponent $c$: | $33$ |
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| Discriminant root field: | $\Q_{2}(\sqrt{-2\cdot 5})$ | |
| 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: | $(6)$ | |
| Jump set: | $[3, 9]$ | |
| Roots of unity: | $14 = (2^{ 3 } - 1) \cdot 2$ |
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Intermediate fields
| 2.3.1.0a1.1, 2.1.3.2a1.1, 2.3.3.6a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 2.3.1.0a1.1 $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{3} + x + 1 \)
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| Relative Eisenstein polynomial: |
\( x^{6} + 4 x^{5} + 4 t^{2} x^{3} + 10 \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^4 + z^2 + 1$,$z + t$ |
| Associated inertia: | $2$,$1$ |
| Indices of inseparability: | $[6, 0]$ |
Invariants of the Galois closure
| Galois degree: | $576$ |
| Galois group: | $A_4^2:C_2^2$ (as 18T176) |
| Inertia group: | Intransitive group isomorphic to $C_2^3\times A_4$ |
| Wild inertia group: | $C_2^5$ |
| Galois unramified degree: | $6$ |
| Galois tame degree: | $3$ |
| Galois Artin slopes: | $[\frac{4}{3}, \frac{4}{3}, 2, 2, 3]$ |
| Galois Swan slopes: | $[\frac{1}{3},\frac{1}{3},1,1,2]$ |
| Galois mean slope: | $2.3958333333333335$ |
| Galois splitting model: |
$x^{18} - 180 x^{14} + 1176 x^{12} + 7776 x^{10} - 42336 x^{8} - 997452 x^{6} + 381024 x^{4} + 6223392 x^{2} - 25412184$
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