Defining polynomial
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$( x^{7} + x + 1 )^{2} + 2 x^{5} ( x^{7} + x + 1 ) + 6$
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $14$ |
| Ramification index $e$: | $2$ |
| Residue field degree $f$: | $7$ |
| Discriminant exponent $c$: | $14$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-1})$ |
| Root number: | $-i$ |
| $\Aut(K/\Q_{2})$: | $C_2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[2]$ |
| Visible Swan slopes: | $[1]$ |
| Means: | $\langle\frac{1}{2}\rangle$ |
| Rams: | $(1)$ |
| Jump set: | $[1, 3]$ |
| Roots of unity: | $254 = (2^{ 7 } - 1) \cdot 2$ |
Intermediate fields
| 2.7.1.0a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 2.7.1.0a1.1 $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{7} + x + 1 \)
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| Relative Eisenstein polynomial: |
\( x^{2} + \left(2 t^{6} + 2 t^{5} + 2 t^{4} + 2 t^{2}\right) x + 2 \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + t^4$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois degree: | $112$ |
| Galois group: | $C_2\times F_8$ (as 14T9) |
| Inertia group: | Intransitive group isomorphic to $C_2^4$ |
| Wild inertia group: | $C_2^4$ |
| Galois unramified degree: | $7$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[2, 2, 2, 2]$ |
| Galois Swan slopes: | $[1,1,1,1]$ |
| Galois mean slope: | $1.875$ |
| Galois splitting model: |
$x^{14} - 84 x^{10} - 70 x^{8} + 1505 x^{6} + 7 x^{4} - 6405 x^{2} + 4489$
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