Defining polynomial
| \( x^{14} + 21 x^{12} + 21 x^{11} - 21 x^{10} + 7 x^{9} - 21 x^{8} - 7 x^{7} + 14 x^{6} - 14 x^{5} + 21 x^{4} - 7 x^{3} - 7 x^{2} - 7 x - 21 \) |
Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$ : | $14$ |
| Ramification exponent $e$ : | $14$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $14$ |
| Discriminant root field: | $\Q_{7}(\sqrt{*})$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 7 })|$: | $1$ |
| This field is not Galois over $\Q_{7}$. | |
Intermediate fields
| $\Q_{7}(\sqrt{7*})$ |
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: | \( x^{14} + 21 x^{12} + 21 x^{11} - 21 x^{10} + 7 x^{9} - 21 x^{8} - 7 x^{7} + 14 x^{6} - 14 x^{5} + 21 x^{4} - 7 x^{3} - 7 x^{2} - 7 x - 21 \) |
Invariants of the Galois closure
| Galois group: | 14T32 |
| Inertia group: | 14T23 |
| Unramified degree: | $2$ |
| Tame degree: | $12$ |
| Wild slopes: | [13/12, 13/12] |
| Galois mean slope: | $635/588$ |
| Galois splitting model: | $x^{14} + 117867955711639 x^{12} - 2266424298073165982752 x^{11} + 33252152864710996966539684552 x^{10} + 428261848017946692118429821298219136 x^{9} + 8983547436911196760849341063186084972677648 x^{8} + 38536050463669986600799818298585805065484951792640 x^{7} + 460555355944260724573373272425560012856391773338659195904 x^{6} + 2933687595003735226503596382915615778809774933424606670743801856 x^{5} + 15391947097878667896844337359748293368970903604944516084841735905839360 x^{4} + 66331268873430834299073134922601563331285857164748145750043014905107796156416 x^{3} + 356187376014069700277092993839122458021977089900994332111163031860895142138689546240 x^{2} + 408763398799804427086135698203190018136432035443173969167926884937720101458279884390301696 x + 3191627607191845984725689029499604666392334154628175148519617536985524305403529354578862704029696$ |