Defining polynomial
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\(x^{14} + 14 x^{13} + 102 x^{12} + 576 x^{11} + 3140 x^{10} + 14392 x^{9} + 49912 x^{8} + 153536 x^{7} + 453296 x^{6} + 957216 x^{5} + 952608 x^{4} + 59648 x^{3} + 31936 x^{2} + 608384 x - 370048\)
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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: | $\Q_{2}(\sqrt{5})$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
| 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: |
\( x^{2} + \left(2 t^{5} + 2 t^{2} + 2\right) x + 4 t^{6} + 4 t^{5} + 4 t + 2 \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + t^{5} + t^{2} + 1$ |
| 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 $C_2^3$ |
| Wild inertia group: | $C_2^3$ |
| Unramified degree: | $14$ |
| Tame degree: | $1$ |
| Wild slopes: | $[2, 2, 2]$ |
| Galois mean slope: | $7/4$ |
| Galois splitting model: |
$x^{14} - 42 x^{12} + 126 x^{10} + 7938 x^{8} - 36855 x^{6} - 304479 x^{4} + 137781 x^{2} + 789507$
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