Defining polynomial
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\(x^{14} - 168 x^{9} + 14 x^{7} - 3528 x^{4} - 7056 x^{3} - 2352 x^{2} + 49\)
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Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $14$ |
| Ramification exponent $e$: | $7$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $14$ |
| Discriminant root field: | $\Q_{7}(\sqrt{3})$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 7 }) }$: | $1$ |
| This field is not Galois over $\Q_{7}.$ | |
| Visible slopes: | $[7/6]$ |
Intermediate fields
| $\Q_{7}(\sqrt{3})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{7}(\sqrt{3})$ $\cong \Q_{7}(t)$ where $t$ is a root of
\( x^{2} + 6 x + 3 \)
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| Relative Eisenstein polynomial: |
\( x^{7} + \left(42 t + 42\right) x^{2} + \left(14 t + 42\right) x + 7 \)
$\ \in\Q_{7}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + 5t + 1$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois group: | $D_7:F_7$ (as 14T24) |
| Inertia group: | Intransitive group isomorphic to $C_7:F_7$ |
| Wild inertia group: | $C_7^2$ |
| Unramified degree: | $2$ |
| Tame degree: | $6$ |
| Wild slopes: | $[7/6, 7/6]$ |
| Galois mean slope: | $341/294$ |
| Galois splitting model: |
$x^{14} - 14 x^{11} - 42 x^{10} - 84 x^{9} - 343 x^{8} - 1392 x^{7} - 4116 x^{6} - 9212 x^{5} - 14952 x^{4} - 17220 x^{3} - 13517 x^{2} - 3234 x + 2844$
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