Defining polynomial
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$( x^{3} + x + 1 )^{4} + 4 x ( x^{3} + x + 1 )^{3} + \left(4 x^{2} + 8\right) ( x^{3} + x + 1 )^{2} + 8 ( x^{3} + x + 1 ) + 10$
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $12$ |
| Ramification index $e$: | $4$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $30$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-5})$ |
| Root number: | $i$ |
| $\Aut(K/\Q_{2})$: | $C_2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[3, \frac{7}{2}]$ |
| Visible Swan slopes: | $[2,\frac{5}{2}]$ |
| Means: | $\langle1, \frac{7}{4}\rangle$ |
| Rams: | $(2, 3)$ |
| Jump set: | $[1, 3, 7]$ |
| Roots of unity: | $14 = (2^{ 3 } - 1) \cdot 2$ |
Intermediate fields
| $\Q_{2}(\sqrt{-2\cdot 5})$, 2.3.1.0a1.1, 2.3.2.9a1.2 |
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: |
\( x^{4} + 4 t x^{3} + 4 t^{2} x^{2} + 10 \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^2 + (t + 1)$,$(t + 1) z + t$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[7, 4, 0]$ |
Invariants of the Galois closure
| Galois degree: | $384$ |
| Galois group: | $C_2\wr C_6$ (as 12T134) |
| Inertia group: | Intransitive group isomorphic to $C_2^3\wr C_2$ |
| Wild inertia group: | $C_2^3\wr C_2$ |
| Galois unramified degree: | $3$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}]$ |
| Galois Swan slopes: | $[1,1,1,2,\frac{5}{2},\frac{5}{2},\frac{5}{2}]$ |
| Galois mean slope: | $3.359375$ |
| Galois splitting model: | $x^{12} - 6 x^{10} - 3 x^{8} + 42 x^{6} + 18 x^{4} - 36 x^{2} + 3$ |