Defining polynomial
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\(x^{14} - 12 x^{13} + 130 x^{12} - 136 x^{11} + 3012 x^{10} + 8480 x^{9} + 51432 x^{8} + 193984 x^{7} + 470000 x^{6} + 1112768 x^{5} + 2842336 x^{4} + 5511552 x^{3} + 6262208 x^{2} + 3639808 x + 686464\)
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $14$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $7$ |
| Discriminant exponent $c$: | $14$ |
| 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: | $[2]$ |
Intermediate fields
| 2.7.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 2.7.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{7} + x + 1 \)
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| Relative Eisenstein polynomial: |
\( x^{2} + \left(2 t^{6} + 2 t^{5} + 2 t^{4} + 2 t^{2}\right) x + 4 t^{6} + 4 t^{5} + 4 t^{3} + 4 t + 6 \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + t^{6} + t^{5} + t^{4} + t^{2}$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_2\times F_8$ (as 14T9) |
| Inertia group: | Intransitive group isomorphic to $C_2^4$ |
| Wild inertia group: | $C_2^4$ |
| Unramified degree: | $7$ |
| Tame degree: | $1$ |
| Wild slopes: | $[2, 2, 2, 2]$ |
| Galois mean slope: | $15/8$ |
| Galois splitting model: |
$x^{14} - 84 x^{10} - 70 x^{8} + 1505 x^{6} + 7 x^{4} - 6405 x^{2} + 4489$
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