Defining polynomial
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\(x^{14} - 154 x^{9} - 112 x^{8} + 14 x^{7} - 1421 x^{4} - 196 x^{3} - 588 x^{2} - 784 x + 49\)
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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: | $\Q_{7}$ |
| 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: |
\( x^{7} + \left(35 t + 28\right) x^{2} + \left(21 t + 7\right) x + 7 \)
$\ \in\Q_{7}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + 4t + 6$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_7^2:C_{12}$ (as 14T23) |
| 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: |
$x^{14} - 8038823564374568007 x^{12} - 11774725679048237237576726986 x^{11} - 731191928246254490365530862004844836 x^{10} + 9892832027708628068999985633391322055990753808 x^{9} + 8235560509565882625005275711024211487559462101021362233 x^{8} + 2520548688490481291699335181763846697601975586081121954072534306 x^{7} + 65217457669075299405706581638387832043113986324244358643026210071282720 x^{6} - 135452801967195685184976240742008265035944258628311149392959345270720028184786088 x^{5} - 26285406081958776406257816513182493645696993640580058769278314615935676860949564122448503 x^{4} - 100305408760340645218538512486922019845876276287141540264789159194769850678392046530856985621282 x^{3} + 429425630783308916268926724184819613175361047610979294132241064707835483922441897546227531290796021823051 x^{2} + 43260130996049873726411770277451068878478186589939713591589596645542021665240667040988878175176218724064080844428 x + 1319569192071309670099844444864615157512505392643493301557702632154693290814991519445878645475148627465759580830153615884$
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