Defining polynomial
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$( x^{6} + x^{4} + x^{3} + x + 1 )^{2} + \left(2 x^{5} + 2 x^{4} + 2 x^{3}\right) ( x^{6} + x^{4} + x^{3} + x + 1 ) + 2$
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $12$ |
| Ramification index $e$: | $2$ |
| Residue field degree $f$: | $6$ |
| Discriminant exponent $c$: | $12$ |
| Discriminant root field: | $\Q_{2}(\sqrt{5})$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{2})$: | $C_2$ |
| 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: | $126 = (2^{ 6 } - 1) \cdot 2$ |
Intermediate fields
| $\Q_{2}(\sqrt{5})$, 2.3.1.0a1.1, 2.6.1.0a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 2.6.1.0a1.1 $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{6} + x^{4} + x^{3} + x + 1 \)
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| Relative Eisenstein polynomial: |
\( x^{2} + \left(2 t^{4} + 2 t^{3} + 2 t^{2} + 2 t\right) x + 4 t + 2 \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + (t^5 + t^4 + t^2 + 1)$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois degree: | $192$ |
| Galois group: | $C_2^4:C_{12}$ (as 12T105) |
| Inertia group: | Intransitive group isomorphic to $C_2^5$ |
| Wild inertia group: | $C_2^5$ |
| Galois unramified degree: | $6$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[2, 2, 2, 2, 2]$ |
| Galois Swan slopes: | $[1,1,1,1,1]$ |
| Galois mean slope: | $1.9375$ |
| Galois splitting model: | $x^{12} - 13 x^{8} + 13 x^{6} + 26 x^{4} - 39 x^{2} + 13$ |