Defining polynomial
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\(x^{12} + 4 x^{11} + 4 x^{8} + 4 x^{7} + 2 x^{6} + 4 x^{5} + 8 x^{3} + 8 x^{2} + 10\)
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $12$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $28$ |
| Discriminant root field: | $\Q_{2}(\sqrt{5})$ |
| Root number: | $-1$ |
| $\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible slopes: | $[2, 10/3]$ |
Intermediate fields
| $\Q_{2}(\sqrt{-1})$, 2.3.2.1, 2.6.8.1 |
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: |
\( x^{12} + 4 x^{11} + 4 x^{8} + 4 x^{7} + 2 x^{6} + 4 x^{5} + 8 x^{3} + 8 x^{2} + 10 \)
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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: | $[17, 6, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_2^4:S_4$ (as 12T145) |
| Inertia group: | $C_2^4:A_4$ (as 12T87) |
| Wild inertia group: | $C_2^3:D_4$ |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | $[2, 2, 8/3, 8/3, 10/3, 10/3]$ |
| Galois mean slope: | $149/48$ |
| Galois splitting model: |
$x^{12} - 20 x^{10} + 15 x^{8} + 50 x^{6} - 55 x^{4} - 90 x^{2} - 27$
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