Defining polynomial
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\(x^{14} + 21 x^{3} + 14 x^{2} + 7\)
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Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $14$ |
| Ramification exponent $e$: | $14$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $15$ |
| Discriminant root field: | $\Q_{7}(\sqrt{7\cdot 3})$ |
| Root number: | $-i$ |
| $\card{ \Aut(K/\Q_{ 7 }) }$: | $2$ |
| This field is not Galois over $\Q_{7}.$ | |
| Visible slopes: | $[7/6]$ |
Intermediate fields
| $\Q_{7}(\sqrt{7\cdot 3})$, 7.7.7.4 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{7}$ |
| Relative Eisenstein polynomial: |
\( x^{14} + 21 x^{3} + 14 x^{2} + 7 \)
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Ramification polygon
| Residual polynomials: | $2z^{2} + 3$,$z^{7} + 2$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[2, 0]$ |
Invariants of the Galois closure
| Galois group: | $F_7$ (as 14T4) |
| Inertia group: | $F_7$ (as 14T4) |
| Wild inertia group: | $C_7$ |
| Unramified degree: | $1$ |
| Tame degree: | $6$ |
| Wild slopes: | $[7/6]$ |
| Galois mean slope: | $47/42$ |
| Galois splitting model: |
$x^{14} - 7 x^{13} + 49 x^{12} - 203 x^{11} + 693 x^{10} - 1771 x^{9} + 3787 x^{8} - 6469 x^{7} + 15323 x^{6} - 28833 x^{5} + 30835 x^{4} - 18585 x^{3} + 6251 x^{2} - 1071 x + 72$
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