Defining polynomial
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\(x^{12} + 8 x^{10} - 2 x^{8} + 6 x^{6} + 2 x^{4} + 4 x^{2} + 6\)
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $12$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $29$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-2\cdot 5})$ |
| Root number: | $-i$ |
| $\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible slopes: | $[2,7/2]$ |
Intermediate fields
| $\Q_{2}(\sqrt{-5})$, 2.3.2.1, 2.6.8.3 |
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: |
\( x^{12} + 8 x^{10} - 2 x^{8} + 6 x^{6} + 2 x^{4} + 4 x^{2} + 6 \)
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| Indices of inseparability: | $[18, 6, 0]$ |
Invariants of the Galois closure
| Galois group: | 12T193 |
| Inertia group: | $C_2^4:C_3.D_4$ (as 12T134) |
| Wild inertia group: | $C_2^3.C_2^4$ |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | $[2, 8/3, 8/3, 3, 10/3, 10/3, 7/2]$ |
| Galois mean slope: | $10/3$ |
| Galois splitting model: |
$x^{12} + 10 x^{10} + 32 x^{8} + 46 x^{6} + 66 x^{4} + 36 x^{2} + 54$
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