Defining polynomial
|
\(x^{14} + 42 x^{2} + 21 x + 7\)
|
Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $14$ |
| Ramification exponent $e$: | $14$ |
| Residue field degree $f$: | $1$ |
| 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: | $[13/12]$ |
Intermediate fields
| $\Q_{7}(\sqrt{7\cdot 3})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{7}$ |
| Relative Eisenstein polynomial: |
\( x^{14} + 42 x^{2} + 21 x + 7 \)
|
Ramification polygon
| Residual polynomials: | $2z + 4$,$z^{7} + 2$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois group: | $D_7^2:C_6$ (as 14T32) |
| Inertia group: | $C_7^2:C_{12}$ (as 14T23) |
| Wild inertia group: | $C_7^2$ |
| Unramified degree: | $2$ |
| Tame degree: | $12$ |
| Wild slopes: | $[13/12, 13/12]$ |
| Galois mean slope: | $635/588$ |
| Galois splitting model: |
$x^{14} + 117867955711639 x^{12} - 2266424298073165982752 x^{11} + 33252152864710996966539684552 x^{10} + 428261848017946692118429821298219136 x^{9} + 8983547436911196760849341063186084972677648 x^{8} + 38536050463669986600799818298585805065484951792640 x^{7} + 460555355944260724573373272425560012856391773338659195904 x^{6} + 2933687595003735226503596382915615778809774933424606670743801856 x^{5} + 15391947097878667896844337359748293368970903604944516084841735905839360 x^{4} + 66331268873430834299073134922601563331285857164748145750043014905107796156416 x^{3} + 356187376014069700277092993839122458021977089900994332111163031860895142138689546240 x^{2} + 408763398799804427086135698203190018136432035443173969167926884937720101458279884390301696 x + 3191627607191845984725689029499604666392334154628175148519617536985524305403529354578862704029696$
|