Defining polynomial
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$( x^{4} + x + 1 )^{3} + 2$
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $12$ |
| Ramification index $e$: | $3$ |
| Residue field degree $f$: | $4$ |
| Discriminant exponent $c$: | $8$ |
| Discriminant root field: | $\Q_{2}(\sqrt{5})$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{2})$ $=$$\Gal(K/\Q_{2})$: | $C_3:C_4$ |
| This field is Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | $[3]$ |
| Roots of unity: | $30 = (2^{ 4 } - 1) \cdot 2$ |
Intermediate fields
| $\Q_{2}(\sqrt{5})$, 2.1.3.2a1.1 x3, 2.4.1.0a1.1, 2.2.3.4a1.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 2.4.1.0a1.1 $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{4} + x + 1 \)
|
| Relative Eisenstein polynomial: |
\( x^{3} + 2 \)
$\ \in\Q_{2}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^{2} + z + 1$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $12$ |
| Galois group: | $C_3:C_4$ (as 12T5) |
| Inertia group: | Intransitive group isomorphic to $C_3$ |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $4$ |
| Galois tame degree: | $3$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[]$ |
| Galois mean slope: | $0.6666666666666666$ |
| Galois splitting model: |
$x^{12} - 3 x^{11} - 888 x^{10} + 2660 x^{9} + 308151 x^{8} - 878220 x^{7} - 53209587 x^{6} + 132388596 x^{5} + 4800412971 x^{4} - 8975649010 x^{3} - 213443493450 x^{2} + 211625342025 x + 3644274033955$
|