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 128

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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
2.1.20.42a1.1	$x^{20} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.2	$x^{20} + 4 x^{10} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.3	$x^{20} + 4 x^{9} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.4	$x^{20} + 4 x^{10} + 4 x^{9} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.5	$x^{20} + 4 x^{7} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.6	$x^{20} + 4 x^{10} + 4 x^{7} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.7	$x^{20} + 4 x^{9} + 4 x^{7} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.8	$x^{20} + 4 x^{10} + 4 x^{9} + 4 x^{7} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.9	$x^{20} + 4 x^{6} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.10	$x^{20} + 4 x^{10} + 4 x^{6} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.11	$x^{20} + 4 x^{9} + 4 x^{6} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.12	$x^{20} + 4 x^{10} + 4 x^{9} + 4 x^{6} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.13	$x^{20} + 4 x^{7} + 4 x^{6} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.14	$x^{20} + 4 x^{10} + 4 x^{7} + 4 x^{6} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.15	$x^{20} + 4 x^{9} + 4 x^{7} + 4 x^{6} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.16	$x^{20} + 4 x^{10} + 4 x^{9} + 4 x^{7} + 4 x^{6} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.17	$x^{20} + 4 x^{5} + 4 x^{3} + 2$	$F_{16}:C_4$ (as 20T173)	$960$	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.18	$x^{20} + 4 x^{10} + 4 x^{5} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.19	$x^{20} + 4 x^{9} + 4 x^{5} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.20	$x^{20} + 4 x^{10} + 4 x^{9} + 4 x^{5} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.21	$x^{20} + 4 x^{7} + 4 x^{5} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.22	$x^{20} + 4 x^{10} + 4 x^{7} + 4 x^{5} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.23	$x^{20} + 4 x^{9} + 4 x^{7} + 4 x^{5} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.24	$x^{20} + 4 x^{10} + 4 x^{9} + 4 x^{7} + 4 x^{5} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.25	$x^{20} + 4 x^{6} + 4 x^{5} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.26	$x^{20} + 4 x^{10} + 4 x^{6} + 4 x^{5} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.27	$x^{20} + 4 x^{9} + 4 x^{6} + 4 x^{5} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.28	$x^{20} + 4 x^{10} + 4 x^{9} + 4 x^{6} + 4 x^{5} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.29	$x^{20} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.30	$x^{20} + 4 x^{10} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.31	$x^{20} + 4 x^{9} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.32	$x^{20} + 4 x^{10} + 4 x^{9} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.33	$x^{20} + 4 x^{3} + 4 x^{2} + 2$	$C_2^4.F_{16}:C_4$ (as 20T472)	$15360$	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.34	$x^{20} + 4 x^{10} + 4 x^{3} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.35	$x^{20} + 4 x^{9} + 4 x^{3} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.36	$x^{20} + 4 x^{10} + 4 x^{9} + 4 x^{3} + 4 x^{2} + 2$	$C_2^6.F_{16}:C_4$ (as 20T683)	$61440$	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.37	$x^{20} + 4 x^{7} + 4 x^{3} + 4 x^{2} + 2$	$C_2^4.F_{16}:C_4$ (as 20T472)	$15360$	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.38	$x^{20} + 4 x^{10} + 4 x^{7} + 4 x^{3} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.39	$x^{20} + 4 x^{9} + 4 x^{7} + 4 x^{3} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.40	$x^{20} + 4 x^{10} + 4 x^{9} + 4 x^{7} + 4 x^{3} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.41	$x^{20} + 4 x^{6} + 4 x^{3} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.42	$x^{20} + 4 x^{10} + 4 x^{6} + 4 x^{3} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.43	$x^{20} + 4 x^{9} + 4 x^{6} + 4 x^{3} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.44	$x^{20} + 4 x^{10} + 4 x^{9} + 4 x^{6} + 4 x^{3} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.45	$x^{20} + 4 x^{7} + 4 x^{6} + 4 x^{3} + 4 x^{2} + 2$	$C_2^4.F_{16}:C_4$ (as 20T472)	$15360$	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.46	$x^{20} + 4 x^{10} + 4 x^{7} + 4 x^{6} + 4 x^{3} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.47	$x^{20} + 4 x^{9} + 4 x^{7} + 4 x^{6} + 4 x^{3} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.48	$x^{20} + 4 x^{10} + 4 x^{9} + 4 x^{7} + 4 x^{6} + 4 x^{3} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.49	$x^{20} + 4 x^{5} + 4 x^{3} + 4 x^{2} + 2$	$C_2^4.F_{16}:C_4$ (as 20T472)	$15360$	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.42a1.50	$x^{20} + 4 x^{10} + 4 x^{5} + 4 x^{3} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 20, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$

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.