Defining polynomial
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\(x^{14} + 35 x^{6} + 21\)
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Invariants
| Base field: | $\Q_{7}$ |
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| Degree $d$: | $14$ |
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| Ramification index $e$: | $14$ |
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| Residue field degree $f$: | $1$ |
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| Discriminant exponent $c$: | $19$ |
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| Discriminant root field: | $\Q_{7}(\sqrt{7})$ | |
| Root number: | $-i$ | |
| $\Aut(K/\Q_{7})$: | $C_2$ | |
| This field is not Galois over $\Q_{7}.$ | ||
| Visible Artin slopes: | $[\frac{3}{2}]$ | |
| Visible Swan slopes: | $[\frac{1}{2}]$ | |
| Means: | $\langle\frac{3}{7}\rangle$ | |
| Rams: | $(1)$ | |
| Jump set: | undefined | |
| Roots of unity: | $6 = (7 - 1)$ |
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Intermediate fields
| $\Q_{7}(\sqrt{7})$, 7.1.7.9a2.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{7}$ |
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| Relative Eisenstein polynomial: |
\( x^{14} + 35 x^{6} + 21 \)
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Ramification polygon
| Residual polynomials: | $z^7 + 2$,$2 z^6 + 4$ |
| Associated inertia: | $1$,$6$ |
| Indices of inseparability: | $[6, 0]$ |
Invariants of the Galois closure
| Galois degree: | $84$ |
| Galois group: | $C_2\times F_7$ (as 14T7) |
| Inertia group: | $D_7$ (as 14T2) |
| Wild inertia group: | $C_7$ |
| Galois unramified degree: | $6$ |
| Galois tame degree: | $2$ |
| Galois Artin slopes: | $[\frac{3}{2}]$ |
| Galois Swan slopes: | $[\frac{1}{2}]$ |
| Galois mean slope: | $1.3571428571428572$ |
| Galois splitting model: |
$x^{14} - 112 x^{12} + 4592 x^{10} - 84448 x^{8} + 720832 x^{6} - 2063488 x^{4} - 75264 x^{2} - 896$
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