Defining polynomial
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$( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 8 x + 2$
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $16$ |
| Ramification index $e$: | $8$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $40$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{2})$: | $C_4$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[\frac{8}{3}, \frac{8}{3}, 3]$ |
| Visible Swan slopes: | $[\frac{5}{3},\frac{5}{3},2]$ |
| Means: | $\langle\frac{5}{6}, \frac{5}{4}, \frac{13}{8}\rangle$ |
| Rams: | $(\frac{5}{3}, \frac{5}{3}, 3)$ |
| Jump set: | $[1, 3, 7, 15]$ |
| Roots of unity: | $6 = (2^{ 2 } - 1) \cdot 2$ |
Intermediate fields
| $\Q_{2}(\sqrt{5})$, 2.2.2.6a1.2, 2.1.4.8a1.1, 2.2.4.16a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{2}(\sqrt{5})$ $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{2} + x + 1 \)
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| Relative Eisenstein polynomial: |
\( x^{8} + 4 x^{7} + 4 x^{5} + 4 x^{4} + 4 x^{2} + 8 t + 2 \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^2 + 1$,$z + 1$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[13, 10, 8, 0]$ |
Invariants of the Galois closure
| Galois degree: | $48$ |
| Galois group: | $A_4:C_4$ (as 16T62) |
| Inertia group: | Intransitive group isomorphic to $C_2\times A_4$ |
| Wild inertia group: | $C_2^3$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $3$ |
| Galois Artin slopes: | $[\frac{8}{3}, \frac{8}{3}, 3]$ |
| Galois Swan slopes: | $[\frac{5}{3},\frac{5}{3},2]$ |
| Galois mean slope: | $2.5833333333333335$ |
| Galois splitting model: |
$x^{16} - 8 x^{15} - 144 x^{14} + 684 x^{13} + 8534 x^{12} - 15540 x^{11} - 224124 x^{10} + 59028 x^{9} + 2683719 x^{8} + 936932 x^{7} - 15745228 x^{6} - 8166244 x^{5} + 45008826 x^{4} + 20359996 x^{3} - 55917608 x^{2} - 15263136 x + 19992411$
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