LMFDB - $p$-adic family 7.1.3.2a1.2-1.7.23a

Defining polynomial

$x^{7} + b_{19} \pi^{3} x^{5} + b_{18} \pi^{3} x^{4} + a_{17} \pi^{3} x^{3} + \pi$

Invariants

Residue field characteristic:	$7$
Degree:	$7$
Base field:	7.1.3.2a1.2
Ramification index $e$:	$7$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$23$
Absolute Artin slopes:	$[\frac{35}{18}]$
Swan slopes:	$[\frac{17}{6}]$
Means:	$\langle\frac{17}{7}\rangle$
Rams:	$(\frac{17}{6})$
Field count:	$98$ (complete)
Ambiguity:	$1$
Mass:	$294$
Absolute Mass:	$98$

Diagrams

Varying

These invariants are all associated to absolute extensions of $\Q_{ 7 }$ within this relative family, not the relative extension.

Indices of inseparability:	$[17,0]$
Associated inertia:	$[1,1]$
Jump Set:	undefined

Fields

Galois group	Galois Artin	Indices

Assoc. Inertia	Jump Set

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Showing 1-50 of 98

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Label	Polynomial $/ \Q_p$	Galois group $/ \Q_p$	Galois degree $/ \Q_p$	$\#\Aut(K/\Q_p)$	Artin slope content $/ \Q_p$	Swan slope content $/ \Q_p$	Hidden Artin slopes $/ \Q_p$	Hidden Swan slopes $/ \Q_p$	Ind. of Insep. $/ \Q_p$	Assoc. Inertia $/ \Q_p$	Resid. Poly	Jump Set
7.1.21.37a1.99	$x^{21} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.100	$x^{21} + 7 x^{19} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.101	$x^{21} + 14 x^{19} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.102	$x^{21} + 21 x^{19} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.103	$x^{21} + 28 x^{19} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.104	$x^{21} + 35 x^{19} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.105	$x^{21} + 42 x^{19} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.106	$x^{21} + 7 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.107	$x^{21} + 7 x^{19} + 7 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.108	$x^{21} + 14 x^{19} + 7 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.109	$x^{21} + 21 x^{19} + 7 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.110	$x^{21} + 28 x^{19} + 7 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.111	$x^{21} + 35 x^{19} + 7 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.112	$x^{21} + 42 x^{19} + 7 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.113	$x^{21} + 14 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.114	$x^{21} + 7 x^{19} + 14 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.115	$x^{21} + 14 x^{19} + 14 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.116	$x^{21} + 21 x^{19} + 14 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.117	$x^{21} + 28 x^{19} + 14 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.118	$x^{21} + 35 x^{19} + 14 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.119	$x^{21} + 42 x^{19} + 14 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.120	$x^{21} + 21 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.121	$x^{21} + 7 x^{19} + 21 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.122	$x^{21} + 14 x^{19} + 21 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.123	$x^{21} + 21 x^{19} + 21 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.124	$x^{21} + 28 x^{19} + 21 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.125	$x^{21} + 35 x^{19} + 21 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.126	$x^{21} + 42 x^{19} + 21 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.127	$x^{21} + 28 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.128	$x^{21} + 7 x^{19} + 28 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.129	$x^{21} + 14 x^{19} + 28 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.130	$x^{21} + 21 x^{19} + 28 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.131	$x^{21} + 28 x^{19} + 28 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.132	$x^{21} + 35 x^{19} + 28 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.133	$x^{21} + 42 x^{19} + 28 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.134	$x^{21} + 35 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.135	$x^{21} + 7 x^{19} + 35 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.136	$x^{21} + 14 x^{19} + 35 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.137	$x^{21} + 21 x^{19} + 35 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.138	$x^{21} + 28 x^{19} + 35 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.139	$x^{21} + 35 x^{19} + 35 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.140	$x^{21} + 42 x^{19} + 35 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.141	$x^{21} + 42 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.142	$x^{21} + 7 x^{19} + 42 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.143	$x^{21} + 14 x^{19} + 42 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.144	$x^{21} + 21 x^{19} + 42 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.145	$x^{21} + 28 x^{19} + 42 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.146	$x^{21} + 35 x^{19} + 42 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.147	$x^{21} + 42 x^{19} + 42 x^{18} + 7 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.37a1.148	$x^{21} + 21 x^{17} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 6$	undefined

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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	7.1.3.2a1.2-1.7.23a
Base	7.1.3.2a1.2
Degree	\(7\)
e	\(7\)
f	\(1\)
c	\(23\)