Defining polynomial
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\(x^{12} + 11 x^{9} + 3 x^{8} - 9 x^{6} - 90 x^{5} + 3 x^{4} - 27 x^{3} + 135 x^{2} + 27 x + 55\)
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $3$ |
| Residue field degree $f$: | $4$ |
| Discriminant exponent $c$: | $8$ |
| Discriminant root field: | $\Q_{2}(\sqrt{5})$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 2 }) }$: | $12$ |
| This field is Galois over $\Q_{2}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{2}(\sqrt{5})$, 2.3.2.1 x3, 2.4.0.1, 2.6.4.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 2.4.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{4} + x + 1 \)
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| Relative Eisenstein polynomial: |
\( x^{3} + 2 \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^{2} + z + 1$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_3:C_4$ (as 12T5) |
| Inertia group: | Intransitive group isomorphic to $C_3$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $4$ |
| Tame degree: | $3$ |
| Wild slopes: | None |
| Galois mean slope: | $2/3$ |
| Galois splitting model: |
$x^{12} - 3 x^{11} - 888 x^{10} + 2660 x^{9} + 308151 x^{8} - 878220 x^{7} - 53209587 x^{6} + 132388596 x^{5} + 4800412971 x^{4} - 8975649010 x^{3} - 213443493450 x^{2} + 211625342025 x + 3644274033955$
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