Defining polynomial
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\(x^{14} + 7 x^{13} + 28 x^{12} + 77 x^{11} + 161 x^{10} + 266 x^{9} + 357 x^{8} + 397 x^{7} + 371 x^{6} + 224 x^{5} + 21 x^{4} + 7 x^{3} + 70 x^{2} + 35 x + 7\)
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $14$ |
| Ramification exponent $e$: | $7$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $12$ |
| 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: | None |
Intermediate fields
| $\Q_{2}(\sqrt{5})$, 2.7.6.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{2}(\sqrt{5})$ $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{2} + x + 1 \)
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| Relative Eisenstein polynomial: |
\( x^{7} + 2 \)
$\ \in\Q_{2}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^{6} + z^{5} + z^{4} + z^{3} + z^{2} + z + 1$ |
| Associated inertia: | $3$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_7:C_6$ (as 14T5) |
| Inertia group: | Intransitive group isomorphic to $C_7$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $6$ |
| Tame degree: | $7$ |
| Wild slopes: | None |
| Galois mean slope: | $6/7$ |
| Galois splitting model: |
$x^{14} + 14 x^{12} + 140 x^{10} + 672 x^{8} - 22 x^{7} + 2352 x^{6} + 308 x^{5} + 3136 x^{4} + 2464 x^{3} + 3136 x^{2} + 1232 x + 484$
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