Defining polynomial
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\(x^{12} + 4 x^{11} + 4 x^{9} + 8 x^{6} + 8 x^{4} + 8 x + 10\)
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $12$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $32$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-1})$ |
| Root number: | $-i$ |
| $\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible slopes: | $[3, 7/2]$ |
Intermediate fields
| 2.3.2.1, 2.6.11.6 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{2}$ |
| Relative Eisenstein polynomial: |
\( x^{12} + 4 x^{11} + 4 x^{9} + 8 x^{6} + 8 x^{4} + 8 x + 10 \)
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Ramification polygon
| Residual polynomials: | $z + 1$,$z^{2} + 1$,$z^{8} + z^{4} + 1$ |
| Associated inertia: | $1$,$1$,$2$ |
| Indices of inseparability: | $[21, 12, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_4^3:D_6$ (as 12T185) |
| Inertia group: | $C_4^3:C_6$ (as 12T141) |
| Wild inertia group: | $C_4^3:C_2$ |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | $[4/3, 4/3, 2, 3, 19/6, 19/6, 7/2]$ |
| Galois mean slope: | $619/192$ |
| Galois splitting model: |
$x^{12} + 54 x^{10} + 909 x^{8} + 2880 x^{6} - 63225 x^{4} - 573750 x^{2} - 1404225$
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