Defining polynomial
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\(x^{12} + 4 x^{11} + 2 x^{10} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x^{2} + 2\)
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $12$ |
| Ramification index $e$: | $12$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $28$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{2})$: | $C_2^2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[\frac{8}{3}, 3]$ |
| Visible Swan slopes: | $[\frac{5}{3},2]$ |
| Means: | $\langle\frac{5}{6}, \frac{17}{12}\rangle$ |
| Rams: | $(5, 7)$ |
| Jump set: | $[3, 9, 21]$ |
| Roots of unity: | $2$ |
Intermediate fields
| $\Q_{2}(\sqrt{2})$, 2.1.3.2a1.1, 2.1.6.10a1.3, 2.1.6.11a1.5, 2.1.6.11a1.14 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{2}$ |
| Relative Eisenstein polynomial: |
\( x^{12} + 4 x^{11} + 2 x^{10} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x^{2} + 2 \)
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Ramification polygon
| Residual polynomials: | $z^8 + z^4 + 1$,$z^2 + 1$,$z + 1$ |
| Associated inertia: | $2$,$1$,$1$ |
| Indices of inseparability: | $[17, 10, 0]$ |
Invariants of the Galois closure
| Galois degree: | $96$ |
| Galois group: | $C_2^2\times S_4$ (as 12T48) |
| Inertia group: | $C_2^2\times A_4$ (as 12T25) |
| Wild inertia group: | $C_2^4$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $3$ |
| Galois Artin slopes: | $[2, \frac{8}{3}, \frac{8}{3}, 3]$ |
| Galois Swan slopes: | $[1,\frac{5}{3},\frac{5}{3},2]$ |
| Galois mean slope: | $2.6666666666666665$ |
| Galois splitting model: | $x^{12} - 4 x^{11} + 16 x^{9} - 11 x^{8} - 28 x^{7} + 32 x^{6} - 16 x^{5} - 33 x^{4} + 64 x^{3} - 20 x^{2} + 32 x + 1$ |