Properties

Label 7.14.14.30
Base \(\Q_{7}\)
Degree \(14\)
e \(14\)
f \(1\)
c \(14\)
Galois group 14T32

Related objects

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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:$x^{14} + 117867955711639 x^{12} - 2266424298073165982752 x^{11} + 33252152864710996966539684552 x^{10} + 428261848017946692118429821298219136 x^{9} + 8983547436911196760849341063186084972677648 x^{8} + 38536050463669986600799818298585805065484951792640 x^{7} + 460555355944260724573373272425560012856391773338659195904 x^{6} + 2933687595003735226503596382915615778809774933424606670743801856 x^{5} + 15391947097878667896844337359748293368970903604944516084841735905839360 x^{4} + 66331268873430834299073134922601563331285857164748145750043014905107796156416 x^{3} + 356187376014069700277092993839122458021977089900994332111163031860895142138689546240 x^{2} + 408763398799804427086135698203190018136432035443173969167926884937720101458279884390301696 x + 3191627607191845984725689029499604666392334154628175148519617536985524305403529354578862704029696$