LMFDB - $p$-adic family 3.1.21.36a

Defining polynomial

$x^{21} + 3 b_{20} x^{20} + 3 b_{19} x^{19} + 3 b_{17} x^{17} + 3 a_{16} x^{16} + 9 c_{24} x^{3} + 9 b_{23} x^{2} + 9 b_{22} x + 3$

Invariants

Residue field characteristic:	$3$
Degree:	$21$
Base field:	$\Q_{3}$
Ramification index $e$:	$21$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$36$
Artin slopes:	$[\frac{15}{7}]$
Swan slopes:	$[\frac{8}{7}]$
Means:	$\langle\frac{16}{21}\rangle$
Rams:	$(8)$
Field count:	$972$ (complete)
Ambiguity:	$3$
Mass:	$486$
Absolute Mass:	$486$

Diagrams

Varying

Indices of inseparability:	$[16,0]$
Associated inertia:	$[6,1]$ (show 729), $[6,2]$ (show 243)
Jump Set:	undefined

Fields

Galois group	Galois Artin	Indices

Assoc. Inertia	Jump Set

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Showing 1-50 of 972

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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
3.1.21.36a1.1	$x^{21} + 6 x^{16} + 3$	$C_3^6:F_7$ (as 21T78)	$30618$	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.2	$x^{21} + 6 x^{16} + 9 x^{2} + 3$	$C_3^6:F_7$ (as 21T78)	$30618$	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.3	$x^{21} + 6 x^{16} + 18 x^{2} + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.4	$x^{21} + 6 x^{16} + 9 x + 3$	$C_3^6:F_7$ (as 21T78)	$30618$	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.5	$x^{21} + 6 x^{16} + 9 x^{2} + 9 x + 3$	$C_3^6:F_7$ (as 21T78)	$30618$	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.6	$x^{21} + 6 x^{16} + 18 x^{2} + 9 x + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.7	$x^{21} + 6 x^{16} + 18 x + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.8	$x^{21} + 6 x^{16} + 9 x^{2} + 18 x + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.9	$x^{21} + 6 x^{16} + 18 x^{2} + 18 x + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.10	$x^{21} + 3 x^{20} + 6 x^{16} + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.11	$x^{21} + 3 x^{20} + 6 x^{16} + 9 x^{2} + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.12	$x^{21} + 3 x^{20} + 6 x^{16} + 18 x^{2} + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.13	$x^{21} + 3 x^{20} + 6 x^{16} + 9 x + 3$	$C_3^6:F_7$ (as 21T78)	$30618$	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.14	$x^{21} + 3 x^{20} + 6 x^{16} + 9 x^{2} + 9 x + 3$	$C_3^6:F_7$ (as 21T78)	$30618$	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.15	$x^{21} + 3 x^{20} + 6 x^{16} + 18 x^{2} + 9 x + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.16	$x^{21} + 3 x^{20} + 6 x^{16} + 18 x + 3$	$C_3^6:F_7$ (as 21T78)	$30618$	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.17	$x^{21} + 3 x^{20} + 6 x^{16} + 9 x^{2} + 18 x + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.18	$x^{21} + 3 x^{20} + 6 x^{16} + 18 x^{2} + 18 x + 3$	$C_3^6:F_7$ (as 21T78)	$30618$	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.19	$x^{21} + 6 x^{20} + 6 x^{16} + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.20	$x^{21} + 6 x^{20} + 6 x^{16} + 9 x^{2} + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.21	$x^{21} + 6 x^{20} + 6 x^{16} + 18 x^{2} + 3$	$C_3^6:F_7$ (as 21T78)	$30618$	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.22	$x^{21} + 6 x^{20} + 6 x^{16} + 9 x + 3$	$C_3^6:F_7$ (as 21T78)	$30618$	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.23	$x^{21} + 6 x^{20} + 6 x^{16} + 9 x^{2} + 9 x + 3$	$C_3^6:F_7$ (as 21T78)	$30618$	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.24	$x^{21} + 6 x^{20} + 6 x^{16} + 18 x^{2} + 9 x + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.25	$x^{21} + 6 x^{20} + 6 x^{16} + 18 x + 3$	$C_3^6:F_7$ (as 21T78)	$30618$	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.26	$x^{21} + 6 x^{20} + 6 x^{16} + 9 x^{2} + 18 x + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.27	$x^{21} + 6 x^{20} + 6 x^{16} + 18 x^{2} + 18 x + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.28	$x^{21} + 3 x^{19} + 6 x^{16} + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.29	$x^{21} + 3 x^{19} + 6 x^{16} + 9 x^{2} + 3$	$C_3^6:F_7$ (as 21T78)	$30618$	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.30	$x^{21} + 3 x^{19} + 6 x^{16} + 18 x^{2} + 3$	$C_3^6:F_7$ (as 21T78)	$30618$	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.31	$x^{21} + 3 x^{19} + 6 x^{16} + 9 x + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.32	$x^{21} + 3 x^{19} + 6 x^{16} + 9 x^{2} + 9 x + 3$	$C_3^6:F_7$ (as 21T78)	$30618$	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.33	$x^{21} + 3 x^{19} + 6 x^{16} + 18 x^{2} + 9 x + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.34	$x^{21} + 3 x^{19} + 6 x^{16} + 18 x + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.35	$x^{21} + 3 x^{19} + 6 x^{16} + 9 x^{2} + 18 x + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.36	$x^{21} + 3 x^{19} + 6 x^{16} + 18 x^{2} + 18 x + 3$	$C_3^6:F_7$ (as 21T78)	$30618$	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.37	$x^{21} + 3 x^{20} + 3 x^{19} + 6 x^{16} + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.38	$x^{21} + 3 x^{20} + 3 x^{19} + 6 x^{16} + 9 x^{2} + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.39	$x^{21} + 3 x^{20} + 3 x^{19} + 6 x^{16} + 18 x^{2} + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.40	$x^{21} + 3 x^{20} + 3 x^{19} + 6 x^{16} + 9 x + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.41	$x^{21} + 3 x^{20} + 3 x^{19} + 6 x^{16} + 9 x^{2} + 9 x + 3$	$C_3^6:F_7$ (as 21T78)	$30618$	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.42	$x^{21} + 3 x^{20} + 3 x^{19} + 6 x^{16} + 18 x^{2} + 9 x + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.43	$x^{21} + 3 x^{20} + 3 x^{19} + 6 x^{16} + 18 x + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.44	$x^{21} + 3 x^{20} + 3 x^{19} + 6 x^{16} + 9 x^{2} + 18 x + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.45	$x^{21} + 3 x^{20} + 3 x^{19} + 6 x^{16} + 18 x^{2} + 18 x + 3$	$C_3^6:F_7$ (as 21T78)	$30618$	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.46	$x^{21} + 6 x^{20} + 3 x^{19} + 6 x^{16} + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.47	$x^{21} + 6 x^{20} + 3 x^{19} + 6 x^{16} + 9 x^{2} + 3$	$C_3^6:F_7$ (as 21T78)	$30618$	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.48	$x^{21} + 6 x^{20} + 3 x^{19} + 6 x^{16} + 18 x^{2} + 3$	$C_3^6:F_7$ (as 21T78)	$30618$	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.49	$x^{21} + 6 x^{20} + 3 x^{19} + 6 x^{16} + 9 x + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined
3.1.21.36a1.50	$x^{21} + 6 x^{20} + 3 x^{19} + 6 x^{16} + 9 x^{2} + 9 x + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[16, 0]$	$[6, 2]$	$z^{18} + z^{15} + 2 z^9 + 2 z^6 + 1,z^2 + 1$	undefined

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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	3.1.21.36a
Base	3.1.1.0a1.1
Degree	\(21\)
e	\(21\)
f	\(1\)
c	\(36\)

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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.