Defining polynomial
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\(x^{12} + 2 x + 2\)
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $12$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $12$ |
| Discriminant root field: | $\Q_{2}(\sqrt{5})$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 2 }) }$: | $1$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible slopes: | $[10/9, 10/9]$ |
Intermediate fields
| 2.3.2.1 |
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: |
\( x^{12} + 2 x + 2 \)
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Ramification polygon
| Residual polynomials: | $z + 1$,$z^{8} + z^{4} + 1$ |
| Associated inertia: | $1$,$2$ |
| Indices of inseparability: | $[1, 1, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_2^6:C_9:C_6$ (as 12T254) |
| Inertia group: | $C_2^6:C_9$ (as 12T166) |
| Wild inertia group: | $C_2^6$ |
| Unramified degree: | $6$ |
| Tame degree: | $9$ |
| Wild slopes: | $[10/9, 10/9, 10/9, 10/9, 10/9, 10/9]$ |
| Galois mean slope: | $319/288$ |
| Galois splitting model: |
$x^{12} - 18 x^{10} - 186 x^{9} + 1179 x^{8} - 1818 x^{7} + 2874 x^{6} - 36360 x^{5} + 192951 x^{4} - 519944 x^{3} + 835722 x^{2} - 786834 x + 359163$
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