LMFDB - $p$-adic family 2.1.20.42a

Defining polynomial

$x^{20} + 2 b_{18} x^{18} + 4 b_{30} x^{10} + 4 b_{29} x^{9} + 4 b_{27} x^{7} + 4 b_{26} x^{6} + 4 b_{25} x^{5} + 4 a_{23} x^{3} + 4 b_{22} x^{2} + 2$

Invariants

Residue field characteristic:	$2$
Degree:	$20$
Base field:	$\Q_{2}$
Ramification index $e$:	$20$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$42$
Artin slopes:	$[\frac{38}{15},\frac{38}{15}]$
Swan slopes:	$[\frac{23}{15},\frac{23}{15}]$
Means:	$\langle\frac{23}{30},\frac{23}{20}\rangle$
Rams:	$(\frac{23}{3},\frac{23}{3})$
Field count:	$128$ (complete)
Ambiguity:	$1$
Mass:	$128$
Absolute Mass:	$128$

Diagrams

Varying

Indices of inseparability:	$[23,18,0]$ (show 64), $[23,20,0]$ (show 64)
Associated inertia:	$[4,1]$
Jump Set:	$[5,15,35]$

Fields

Galois group	Galois Artin	Indices

Assoc. Inertia	Jump Set

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Showing 1-50 of 64

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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
2.1.20.42a1.65	$x^{20} + 2 x^{18} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.66	$x^{20} + 2 x^{18} + 4 x^{10} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.67	$x^{20} + 2 x^{18} + 4 x^{9} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.68	$x^{20} + 2 x^{18} + 4 x^{10} + 4 x^{9} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.69	$x^{20} + 2 x^{18} + 4 x^{7} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.70	$x^{20} + 2 x^{18} + 4 x^{10} + 4 x^{7} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.71	$x^{20} + 2 x^{18} + 4 x^{9} + 4 x^{7} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.72	$x^{20} + 2 x^{18} + 4 x^{10} + 4 x^{9} + 4 x^{7} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.73	$x^{20} + 2 x^{18} + 4 x^{6} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.74	$x^{20} + 2 x^{18} + 4 x^{10} + 4 x^{6} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.75	$x^{20} + 2 x^{18} + 4 x^{9} + 4 x^{6} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.76	$x^{20} + 2 x^{18} + 4 x^{10} + 4 x^{9} + 4 x^{6} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.77	$x^{20} + 2 x^{18} + 4 x^{7} + 4 x^{6} + 4 x^{3} + 2$	$C_2^2:F_{16}:C_4$ (as 20T282)	$3840$	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.78	$x^{20} + 2 x^{18} + 4 x^{10} + 4 x^{7} + 4 x^{6} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.79	$x^{20} + 2 x^{18} + 4 x^{9} + 4 x^{7} + 4 x^{6} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.80	$x^{20} + 2 x^{18} + 4 x^{10} + 4 x^{9} + 4 x^{7} + 4 x^{6} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.81	$x^{20} + 2 x^{18} + 4 x^{5} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.82	$x^{20} + 2 x^{18} + 4 x^{10} + 4 x^{5} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.83	$x^{20} + 2 x^{18} + 4 x^{9} + 4 x^{5} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.84	$x^{20} + 2 x^{18} + 4 x^{10} + 4 x^{9} + 4 x^{5} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.85	$x^{20} + 2 x^{18} + 4 x^{7} + 4 x^{5} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.86	$x^{20} + 2 x^{18} + 4 x^{10} + 4 x^{7} + 4 x^{5} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.87	$x^{20} + 2 x^{18} + 4 x^{9} + 4 x^{7} + 4 x^{5} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.88	$x^{20} + 2 x^{18} + 4 x^{10} + 4 x^{9} + 4 x^{7} + 4 x^{5} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.89	$x^{20} + 2 x^{18} + 4 x^{6} + 4 x^{5} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.90	$x^{20} + 2 x^{18} + 4 x^{10} + 4 x^{6} + 4 x^{5} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.91	$x^{20} + 2 x^{18} + 4 x^{9} + 4 x^{6} + 4 x^{5} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.92	$x^{20} + 2 x^{18} + 4 x^{10} + 4 x^{9} + 4 x^{6} + 4 x^{5} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.93	$x^{20} + 2 x^{18} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.94	$x^{20} + 2 x^{18} + 4 x^{10} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x^{3} + 2$	$C_2^6.F_{16}:C_4$ (as 20T683)	$61440$	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.95	$x^{20} + 2 x^{18} + 4 x^{9} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.96	$x^{20} + 2 x^{18} + 4 x^{10} + 4 x^{9} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.97	$x^{20} + 2 x^{18} + 4 x^{3} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.98	$x^{20} + 2 x^{18} + 4 x^{10} + 4 x^{3} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.99	$x^{20} + 2 x^{18} + 4 x^{9} + 4 x^{3} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.100	$x^{20} + 2 x^{18} + 4 x^{10} + 4 x^{9} + 4 x^{3} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.101	$x^{20} + 2 x^{18} + 4 x^{7} + 4 x^{3} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.102	$x^{20} + 2 x^{18} + 4 x^{10} + 4 x^{7} + 4 x^{3} + 4 x^{2} + 2$	$C_2^4.F_{16}:C_4$ (as 20T472)	$15360$	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.103	$x^{20} + 2 x^{18} + 4 x^{9} + 4 x^{7} + 4 x^{3} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.104	$x^{20} + 2 x^{18} + 4 x^{10} + 4 x^{9} + 4 x^{7} + 4 x^{3} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.105	$x^{20} + 2 x^{18} + 4 x^{6} + 4 x^{3} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.106	$x^{20} + 2 x^{18} + 4 x^{10} + 4 x^{6} + 4 x^{3} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.107	$x^{20} + 2 x^{18} + 4 x^{9} + 4 x^{6} + 4 x^{3} + 4 x^{2} + 2$	$C_2^6.F_{16}:C_4$ (as 20T683)	$61440$	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.108	$x^{20} + 2 x^{18} + 4 x^{10} + 4 x^{9} + 4 x^{6} + 4 x^{3} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.109	$x^{20} + 2 x^{18} + 4 x^{7} + 4 x^{6} + 4 x^{3} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.110	$x^{20} + 2 x^{18} + 4 x^{10} + 4 x^{7} + 4 x^{6} + 4 x^{3} + 4 x^{2} + 2$	$C_2^4.F_{16}:C_4$ (as 20T472)	$15360$	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.111	$x^{20} + 2 x^{18} + 4 x^{9} + 4 x^{7} + 4 x^{6} + 4 x^{3} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.112	$x^{20} + 2 x^{18} + 4 x^{10} + 4 x^{9} + 4 x^{7} + 4 x^{6} + 4 x^{3} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.113	$x^{20} + 2 x^{18} + 4 x^{5} + 4 x^{3} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.114	$x^{20} + 2 x^{18} + 4 x^{10} + 4 x^{5} + 4 x^{3} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 18, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$

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	2.1.20.42a
Base	2.1.1.0a1.1
Degree	\(20\)
e	\(20\)
f	\(1\)
c	\(42\)

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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.