Defining polynomial
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$( x^{3} + x + 1 )^{4} + 2 ( x^{3} + x + 1 )^{3} + 2 ( x^{3} + x + 1 )^{2} + 2$
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $12$ |
| Ramification index $e$: | $4$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $18$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $-1$ |
| $\Aut(K/\Q_{2})$ $=$$\Gal(K/\Q_{2})$: | $A_4$ |
| This field is Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[2, 2]$ |
| Visible Swan slopes: | $[1,1]$ |
| Means: | $\langle\frac{1}{2}, \frac{3}{4}\rangle$ |
| Rams: | $(1, 1)$ |
| Jump set: | $[1, 2, 7]$ |
| Roots of unity: | $14 = (2^{ 3 } - 1) \cdot 2$ |
Intermediate fields
| 2.3.1.0a1.1, 2.1.4.6a2.1 x4, 2.3.2.6a3.1 x3 |
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: |
\( x^{4} + 2 x^{3} + 2 x^{2} + 2 \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^3 + (t^2 + t + 1) z + (t^2 + t)$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[3, 2, 0]$ |
Invariants of the Galois closure
| Galois degree: | $12$ |
| Galois group: | $A_4$ (as 12T4) |
| Inertia group: | Intransitive group isomorphic to $C_2^2$ |
| Wild inertia group: | $C_2^2$ |
| Galois unramified degree: | $3$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[2, 2]$ |
| Galois Swan slopes: | $[1,1]$ |
| Galois mean slope: | $1.5$ |
| Galois splitting model: | $x^{12} - 2 x^{11} - 4 x^{10} + 6 x^{9} + 18 x^{8} - 24 x^{7} - 18 x^{6} + 12 x^{5} + 57 x^{4} - 76 x^{3} + 40 x^{2} - 10 x + 1$ |