LMFDB - $p$-adic family 7.1.21.35a

Defining polynomial

$x^{21} + 7 b_{17} x^{17} + 7 b_{16} x^{16} + 7 a_{15} x^{15} + 7 d_{0}$

Invariants

Residue field characteristic:	$7$
Degree:	$21$
Base field:	$\Q_{7}$
Ramification index $e$:	$21$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$35$
Artin slopes:	$[\frac{11}{6}]$
Swan slopes:	$[\frac{5}{6}]$
Means:	$\langle\frac{5}{7}\rangle$
Rams:	$(\frac{5}{2})$
Field count:	$306$ (complete)
Ambiguity:	$3$
Mass:	$294$
Absolute Mass:	$294$

Diagrams

Varying

Indices of inseparability:	$[15,0]$
Associated inertia:	$[1,1]$ (show 102), $[1,3]$ (show 204)
Jump Set:	undefined

Fields

Galois group	Galois Artin	Indices

Assoc. Inertia	Jump Set

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Showing 1-50 of 306

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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
7.1.21.35a1.1	$x^{21} + 7 x^{15} + 7$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 6$	undefined
7.1.21.35a1.2	$x^{21} + 7 x^{17} + 7 x^{15} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 6$	undefined
7.1.21.35a1.3	$x^{21} + 21 x^{17} + 7 x^{15} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 6$	undefined
7.1.21.35a1.4	$x^{21} + 7 x^{16} + 7 x^{15} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 6$	undefined
7.1.21.35a1.5	$x^{21} + 7 x^{17} + 7 x^{16} + 7 x^{15} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 6$	undefined
7.1.21.35a1.6	$x^{21} + 14 x^{17} + 7 x^{16} + 7 x^{15} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 6$	undefined
7.1.21.35a1.7	$x^{21} + 21 x^{17} + 7 x^{16} + 7 x^{15} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 6$	undefined
7.1.21.35a1.8	$x^{21} + 28 x^{17} + 7 x^{16} + 7 x^{15} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 6$	undefined
7.1.21.35a1.9	$x^{21} + 35 x^{17} + 7 x^{16} + 7 x^{15} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 6$	undefined
7.1.21.35a1.10	$x^{21} + 42 x^{17} + 7 x^{16} + 7 x^{15} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 6$	undefined
7.1.21.35a1.11	$x^{21} + 21 x^{16} + 7 x^{15} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 6$	undefined
7.1.21.35a1.12	$x^{21} + 7 x^{17} + 21 x^{16} + 7 x^{15} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 6$	undefined
7.1.21.35a1.13	$x^{21} + 14 x^{17} + 21 x^{16} + 7 x^{15} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 6$	undefined
7.1.21.35a1.14	$x^{21} + 21 x^{17} + 21 x^{16} + 7 x^{15} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 6$	undefined
7.1.21.35a1.15	$x^{21} + 28 x^{17} + 21 x^{16} + 7 x^{15} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 6$	undefined
7.1.21.35a1.16	$x^{21} + 35 x^{17} + 21 x^{16} + 7 x^{15} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 6$	undefined
7.1.21.35a1.17	$x^{21} + 42 x^{17} + 21 x^{16} + 7 x^{15} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 6$	undefined
7.1.21.35a1.18	$x^{21} + 42 x^{15} + 7$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 1$	undefined
7.1.21.35a1.19	$x^{21} + 7 x^{17} + 42 x^{15} + 7$	$C_7:(C_3\times F_7)$ (as 21T24)	$882$	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 1$	undefined
7.1.21.35a1.20	$x^{21} + 21 x^{17} + 42 x^{15} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 1$	undefined
7.1.21.35a1.21	$x^{21} + 7 x^{16} + 42 x^{15} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 1$	undefined
7.1.21.35a1.22	$x^{21} + 7 x^{17} + 7 x^{16} + 42 x^{15} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 1$	undefined
7.1.21.35a1.23	$x^{21} + 14 x^{17} + 7 x^{16} + 42 x^{15} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 1$	undefined
7.1.21.35a1.24	$x^{21} + 21 x^{17} + 7 x^{16} + 42 x^{15} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 1$	undefined
7.1.21.35a1.25	$x^{21} + 28 x^{17} + 7 x^{16} + 42 x^{15} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 1$	undefined
7.1.21.35a1.26	$x^{21} + 35 x^{17} + 7 x^{16} + 42 x^{15} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 1$	undefined
7.1.21.35a1.27	$x^{21} + 42 x^{17} + 7 x^{16} + 42 x^{15} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 1$	undefined
7.1.21.35a1.28	$x^{21} + 21 x^{16} + 42 x^{15} + 7$	$C_7:(C_3\times F_7)$ (as 21T24)	$882$	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 1$	undefined
7.1.21.35a1.29	$x^{21} + 7 x^{17} + 21 x^{16} + 42 x^{15} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 1$	undefined
7.1.21.35a1.30	$x^{21} + 14 x^{17} + 21 x^{16} + 42 x^{15} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 1$	undefined
7.1.21.35a1.31	$x^{21} + 21 x^{17} + 21 x^{16} + 42 x^{15} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 1$	undefined
7.1.21.35a1.32	$x^{21} + 28 x^{17} + 21 x^{16} + 42 x^{15} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 1$	undefined
7.1.21.35a1.33	$x^{21} + 35 x^{17} + 21 x^{16} + 42 x^{15} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 1$	undefined
7.1.21.35a1.34	$x^{21} + 42 x^{17} + 21 x^{16} + 42 x^{15} + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 1$	undefined
7.1.21.35a1.35	$x^{21} + 14 x^{15} + 14$	$C_3\times F_7$ (as 21T9)	$126$	$3$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 6$	undefined
7.1.21.35a1.36	$x^{21} + 7 x^{17} + 14 x^{15} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 6$	undefined
7.1.21.35a1.37	$x^{21} + 21 x^{17} + 14 x^{15} + 14$	$C_7:(C_3\times F_7)$ (as 21T24)	$882$	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 6$	undefined
7.1.21.35a1.38	$x^{21} + 7 x^{16} + 14 x^{15} + 14$	$C_7:(C_3\times F_7)$ (as 21T24)	$882$	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 6$	undefined
7.1.21.35a1.39	$x^{21} + 7 x^{17} + 7 x^{16} + 14 x^{15} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 6$	undefined
7.1.21.35a1.40	$x^{21} + 14 x^{17} + 7 x^{16} + 14 x^{15} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 6$	undefined
7.1.21.35a1.41	$x^{21} + 21 x^{17} + 7 x^{16} + 14 x^{15} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 6$	undefined
7.1.21.35a1.42	$x^{21} + 28 x^{17} + 7 x^{16} + 14 x^{15} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 6$	undefined
7.1.21.35a1.43	$x^{21} + 35 x^{17} + 7 x^{16} + 14 x^{15} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 6$	undefined
7.1.21.35a1.44	$x^{21} + 42 x^{17} + 7 x^{16} + 14 x^{15} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 6$	undefined
7.1.21.35a1.45	$x^{21} + 21 x^{16} + 14 x^{15} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 6$	undefined
7.1.21.35a1.46	$x^{21} + 7 x^{17} + 21 x^{16} + 14 x^{15} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 6$	undefined
7.1.21.35a1.47	$x^{21} + 14 x^{17} + 21 x^{16} + 14 x^{15} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 6$	undefined
7.1.21.35a1.48	$x^{21} + 21 x^{17} + 21 x^{16} + 14 x^{15} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 6$	undefined
7.1.21.35a1.49	$x^{21} + 28 x^{17} + 21 x^{16} + 14 x^{15} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 6$	undefined
7.1.21.35a1.50	$x^{21} + 35 x^{17} + 21 x^{16} + 14 x^{15} + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[15, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^3 + 6$	undefined

Next Download displayed columns for results to

Overview	Random
Universe	Knowledge

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	7.1.21.35a
Base	7.1.1.0a1.1
Degree	\(21\)
e	\(21\)
f	\(1\)
c	\(35\)

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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.