LMFDB - $p$-adic family 2.1.20.38a

Defining polynomial

$x^{20} + 2 a_{19} x^{19} + 2 b_{18} x^{18} + 2 b_{14} x^{14} + 4 b_{25} x^{5} + 4 b_{23} x^{3} + 4 b_{22} x^{2} + 4 b_{21} x + 2$

Invariants

Residue field characteristic:	$2$
Degree:	$20$
Base field:	$\Q_{2}$
Ramification index $e$:	$20$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$38$
Artin slopes:	$[\frac{34}{15},\frac{34}{15}]$
Swan slopes:	$[\frac{19}{15},\frac{19}{15}]$
Means:	$\langle\frac{19}{30},\frac{19}{20}\rangle$
Rams:	$(\frac{19}{3},\frac{19}{3})$
Field count:	$64$ (complete)
Ambiguity:	$1$
Mass:	$64$
Absolute Mass:	$64$

Diagrams

Varying

Indices of inseparability:	$[19,14,0]$ (show 32), $[19,18,0]$ (show 16), $[19,19,0]$ (show 16)
Associated inertia:	$[4,1]$
Jump Set:	$[5,15,35]$

Fields

Galois group	Galois Artin	Indices

Assoc. Inertia	Jump Set

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Showing all 32

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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
2.1.20.38a1.33	$x^{20} + 2 x^{19} + 2 x^{14} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[19, 14, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.38a1.34	$x^{20} + 2 x^{19} + 2 x^{14} + 4 x^{5} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[19, 14, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.38a1.35	$x^{20} + 2 x^{19} + 2 x^{14} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[19, 14, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.38a1.36	$x^{20} + 2 x^{19} + 2 x^{14} + 4 x^{5} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[19, 14, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.38a1.37	$x^{20} + 2 x^{19} + 2 x^{14} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[19, 14, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.38a1.38	$x^{20} + 2 x^{19} + 2 x^{14} + 4 x^{5} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[19, 14, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.38a1.39	$x^{20} + 2 x^{19} + 2 x^{14} + 4 x^{3} + 4 x^{2} + 2$	$C_2^4.F_{16}:C_4$ (as 20T472)	$15360$	$1$	not computed	not computed	not computed	not computed	$[19, 14, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.38a1.40	$x^{20} + 2 x^{19} + 2 x^{14} + 4 x^{5} + 4 x^{3} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[19, 14, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.38a1.41	$x^{20} + 2 x^{19} + 2 x^{14} + 4 x + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[19, 14, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.38a1.42	$x^{20} + 2 x^{19} + 2 x^{14} + 4 x^{5} + 4 x + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[19, 14, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.38a1.43	$x^{20} + 2 x^{19} + 2 x^{14} + 4 x^{3} + 4 x + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[19, 14, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.38a1.44	$x^{20} + 2 x^{19} + 2 x^{14} + 4 x^{5} + 4 x^{3} + 4 x + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[19, 14, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.38a1.45	$x^{20} + 2 x^{19} + 2 x^{14} + 4 x^{2} + 4 x + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[19, 14, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.38a1.46	$x^{20} + 2 x^{19} + 2 x^{14} + 4 x^{5} + 4 x^{2} + 4 x + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[19, 14, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.38a1.47	$x^{20} + 2 x^{19} + 2 x^{14} + 4 x^{3} + 4 x^{2} + 4 x + 2$	$C_2^4.F_{16}:C_4$ (as 20T472)	$15360$	$1$	not computed	not computed	not computed	not computed	$[19, 14, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.38a1.48	$x^{20} + 2 x^{19} + 2 x^{14} + 4 x^{5} + 4 x^{3} + 4 x^{2} + 4 x + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[19, 14, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.38a1.49	$x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{14} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[19, 14, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.38a1.50	$x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{14} + 4 x^{5} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[19, 14, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.38a1.51	$x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{14} + 4 x^{3} + 2$	$C_2^6.F_{16}:C_4$ (as 20T683)	$61440$	$1$	not computed	not computed	not computed	not computed	$[19, 14, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.38a1.52	$x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{14} + 4 x^{5} + 4 x^{3} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[19, 14, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.38a1.53	$x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{14} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[19, 14, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.38a1.54	$x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{14} + 4 x^{5} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[19, 14, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.38a1.55	$x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{14} + 4 x^{3} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[19, 14, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.38a1.56	$x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{14} + 4 x^{5} + 4 x^{3} + 4 x^{2} + 2$	$C_2^4.F_{16}:C_4$ (as 20T472)	$15360$	$1$	not computed	not computed	not computed	not computed	$[19, 14, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.38a1.57	$x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{14} + 4 x + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[19, 14, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.38a1.58	$x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{14} + 4 x^{5} + 4 x + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[19, 14, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.38a1.59	$x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{14} + 4 x^{3} + 4 x + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[19, 14, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.38a1.60	$x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{14} + 4 x^{5} + 4 x^{3} + 4 x + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[19, 14, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.38a1.61	$x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{14} + 4 x^{2} + 4 x + 2$	$C_2^4.F_{16}:C_4$ (as 20T472)	$15360$	$1$	not computed	not computed	not computed	not computed	$[19, 14, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.38a1.62	$x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{14} + 4 x^{5} + 4 x^{2} + 4 x + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[19, 14, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.38a1.63	$x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{14} + 4 x^{3} + 4 x^{2} + 4 x + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[19, 14, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$
2.1.20.38a1.64	$x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{14} + 4 x^{5} + 4 x^{3} + 4 x^{2} + 4 x + 2$	$F_{16}:C_4$ (as 20T173)	$960$	$1$	not computed	not computed	not computed	not computed	$[19, 14, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 15, 35]$

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	2.1.20.38a
Base	2.1.1.0a1.1
Degree	\(20\)
e	\(20\)
f	\(1\)
c	\(38\)

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Defining polynomial

Invariants

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Varying

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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.