LMFDB - $p$-adic family 7.1.21.37a

Defining polynomial

$x^{21} + 7 b_{19} x^{19} + 7 b_{18} x^{18} + 7 a_{17} x^{17} + 7 d_{0}$

Invariants

Residue field characteristic:	$7$
Degree:	$21$
Base field:	$\Q_{7}$
Ramification index $e$:	$21$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$37$
Artin slopes:	$[\frac{35}{18}]$
Swan slopes:	$[\frac{17}{18}]$
Means:	$\langle\frac{17}{21}\rangle$
Rams:	$(\frac{17}{6})$
Field count:	$294$ (complete)
Ambiguity:	$3$
Mass:	$294$
Absolute Mass:	$294$

Diagrams

Varying

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 294

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Label	Polynomial	Galois group	Galois degree	$\#\Aut(K/\Q_p)$	Artin slope content	Swan slope content	Hidden Artin slopes	Hidden Swan slopes	Ind. of Insep.	Assoc. Inertia	Resid. Poly	Jump Set
7.1.21.37a1.1	$x^{21} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.2	$x^{21} + 7 x^{19} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.3	$x^{21} + 14 x^{19} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.4	$x^{21} + 21 x^{19} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.5	$x^{21} + 28 x^{19} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.6	$x^{21} + 35 x^{19} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.7	$x^{21} + 42 x^{19} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.8	$x^{21} + 7 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.9	$x^{21} + 7 x^{19} + 7 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.10	$x^{21} + 14 x^{19} + 7 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.11	$x^{21} + 21 x^{19} + 7 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.12	$x^{21} + 28 x^{19} + 7 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.13	$x^{21} + 35 x^{19} + 7 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.14	$x^{21} + 42 x^{19} + 7 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.15	$x^{21} + 14 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.16	$x^{21} + 7 x^{19} + 14 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.17	$x^{21} + 14 x^{19} + 14 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.18	$x^{21} + 21 x^{19} + 14 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.19	$x^{21} + 28 x^{19} + 14 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.20	$x^{21} + 35 x^{19} + 14 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.21	$x^{21} + 42 x^{19} + 14 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.22	$x^{21} + 21 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.23	$x^{21} + 7 x^{19} + 21 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.24	$x^{21} + 14 x^{19} + 21 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.25	$x^{21} + 21 x^{19} + 21 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.26	$x^{21} + 28 x^{19} + 21 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.27	$x^{21} + 35 x^{19} + 21 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.28	$x^{21} + 42 x^{19} + 21 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.29	$x^{21} + 28 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.30	$x^{21} + 7 x^{19} + 28 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.31	$x^{21} + 14 x^{19} + 28 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.32	$x^{21} + 21 x^{19} + 28 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.33	$x^{21} + 28 x^{19} + 28 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.34	$x^{21} + 35 x^{19} + 28 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.35	$x^{21} + 42 x^{19} + 28 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.36	$x^{21} + 35 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.37	$x^{21} + 7 x^{19} + 35 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.38	$x^{21} + 14 x^{19} + 35 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.39	$x^{21} + 21 x^{19} + 35 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.40	$x^{21} + 28 x^{19} + 35 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.41	$x^{21} + 35 x^{19} + 35 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.42	$x^{21} + 42 x^{19} + 35 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.43	$x^{21} + 42 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.44	$x^{21} + 7 x^{19} + 42 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.45	$x^{21} + 14 x^{19} + 42 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.46	$x^{21} + 21 x^{19} + 42 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.47	$x^{21} + 28 x^{19} + 42 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.48	$x^{21} + 35 x^{19} + 42 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.49	$x^{21} + 42 x^{19} + 42 x^{18} + 7 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.37a1.50	$x^{21} + 21 x^{17} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[17, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 5$	undefined

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Label	7.1.21.37a
Base	7.1.1.0a1.1
Degree	\(21\)
e	\(21\)
f	\(1\)
c	\(37\)

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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.