Defining polynomial
$( x^{3} + x + 1 )^{4} + 8 ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + 8 x^{2} ( x^{3} + x + 1 ) + 2$
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Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $12$ |
Ramification index $e$: | $4$ |
Residue field degree $f$: | $3$ |
Discriminant exponent $c$: | $33$ |
Discriminant root field: | $\Q_{2}(\sqrt{2})$ |
Root number: | $1$ |
$\Aut(K/\Q_{2})$: | $C_2$ |
This field is not Galois over $\Q_{2}.$ | |
Visible Artin slopes: | $[3, 4]$ |
Visible Swan slopes: | $[2,3]$ |
Means: | $\langle1, 2\rangle$ |
Rams: | $(2, 4)$ |
Jump set: | $[1, 3, 7]$ |
Roots of unity: | $14 = (2^{ 3 } - 1) \cdot 2$ |
Intermediate fields
$\Q_{2}(\sqrt{2})$, 2.3.1.0a1.1, 2.3.2.9a1.5 |
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: |
\( x^{4} + 8 x^{3} + 4 x^{2} + 8 t x + 2 \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
Residual polynomials: | $z^2 + (t + 1)$,$(t + 1) z + (t^2 + 1)$ |
Associated inertia: | $1$,$1$ |
Indices of inseparability: | $[8, 4, 0]$ |
Invariants of the Galois closure
Galois degree: | $192$ |
Galois group: | $C_2^4:C_{12}$ (as 12T105) |
Inertia group: | Intransitive group isomorphic to $C_2^4:C_4$ |
Wild inertia group: | $C_2^4:C_4$ |
Galois unramified degree: | $3$ |
Galois tame degree: | $1$ |
Galois Artin slopes: | $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]$ |
Galois Swan slopes: | $[1,1,2,\frac{5}{2},\frac{5}{2},3]$ |
Galois mean slope: | $3.59375$ |
Galois splitting model: | $x^{12} + 12 x^{10} + 40 x^{8} - 172 x^{4} - 176 x^{2} + 8$ |