$p$-adic family 2.1.10.19a1.61-1.2.21a

• Label: 2.1.10.19a1.61-1.2.21a
• Base: 2.1.10.19a1.61
• Degree: \(2\)
• e: \(2\)
• f: \(1\)
• c: \(21\)

Defining polynomial

$x^{2} + \left(b_{39} \pi^{20} + b_{37} \pi^{19} + b_{35} \pi^{18} + b_{33} \pi^{17} + b_{31} \pi^{16} + b_{29} \pi^{15} + b_{27} \pi^{14} + b_{25} \pi^{13} + b_{23} \pi^{12} + b_{21} \pi^{11}\right) x + c_{40} \pi^{21} + \pi$

Invariants

Residue field characteristic:	$2$
Degree:	$2$
Base field:	2.1.10.19a1.61
Ramification index $e$:	$2$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$21$
Absolute Artin slopes:	$[3,4]$
Swan slopes:	$[20]$
Means:	$\langle10\rangle$
Rams:	$(20)$
Field count:	$1056$ (complete)
Ambiguity:	$2$
Mass:	$1024$
Absolute Mass:	$512$

Diagrams

Varying

These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.

Indices of inseparability:	$[40,20,0]$
Associated inertia:	$[4,1,1]$
Jump Set:	$[5,15,35]$

Fields

Showing 1-50 of

  To download results, . Download displayed columns for results to

Label	Polynomial $/ \Q_p$	Galois group $/ \Q_p$	Galois degree $/ \Q_p$	$\#\Aut(K/\Q_p)$	Artin slope content $/ \Q_p$	Swan slope content $/ \Q_p$	Hidden Artin slopes $/ \Q_p$	Hidden Swan slopes $/ \Q_p$	Ind. of Insep. $/ \Q_p$	Assoc. Inertia $/ \Q_p$	Resid. Poly	Jump Set
2.1.20.59a1.62337	$x^{20} + 4 x^{14} + 4 x^{10} + 4 x^{6} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62338	$x^{20} + 4 x^{14} + 4 x^{10} + 4 x^{6} + 4 x^{2} + 18$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62339	$x^{20} + 8 x^{19} + 4 x^{14} + 4 x^{10} + 4 x^{6} + 4 x^{2} + 2$	$C_2^8.(C_4\times F_5)$ (as 20T523)	$20480$	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62340	$x^{20} + 8 x^{19} + 4 x^{14} + 4 x^{10} + 4 x^{6} + 4 x^{2} + 18$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62341	$x^{20} + 8 x^{17} + 4 x^{14} + 4 x^{10} + 4 x^{6} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62342	$x^{20} + 8 x^{17} + 4 x^{14} + 4 x^{10} + 4 x^{6} + 4 x^{2} + 18$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62343	$x^{20} + 8 x^{19} + 8 x^{17} + 4 x^{14} + 4 x^{10} + 4 x^{6} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62344	$x^{20} + 8 x^{19} + 8 x^{17} + 4 x^{14} + 4 x^{10} + 4 x^{6} + 4 x^{2} + 18$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62345	$x^{20} + 8 x^{15} + 4 x^{14} + 4 x^{10} + 4 x^{6} + 4 x^{2} + 2$	$C_2^8.(C_4\times F_5)$ (as 20T523)	$20480$	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62346	$x^{20} + 8 x^{15} + 4 x^{14} + 4 x^{10} + 4 x^{6} + 4 x^{2} + 18$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62347	$x^{20} + 8 x^{19} + 8 x^{15} + 4 x^{14} + 4 x^{10} + 4 x^{6} + 4 x^{2} + 2$	$C_2^8.(C_4\times F_5)$ (as 20T523)	$20480$	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62348	$x^{20} + 8 x^{19} + 8 x^{15} + 4 x^{14} + 4 x^{10} + 4 x^{6} + 4 x^{2} + 18$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62349	$x^{20} + 8 x^{17} + 8 x^{15} + 4 x^{14} + 4 x^{10} + 4 x^{6} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62350	$x^{20} + 8 x^{17} + 8 x^{15} + 4 x^{14} + 4 x^{10} + 4 x^{6} + 4 x^{2} + 18$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62351	$x^{20} + 8 x^{19} + 8 x^{17} + 8 x^{15} + 4 x^{14} + 4 x^{10} + 4 x^{6} + 4 x^{2} + 2$	$C_2^8.(C_4\times F_5)$ (as 20T523)	$20480$	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62352	$x^{20} + 8 x^{19} + 8 x^{17} + 8 x^{15} + 4 x^{14} + 4 x^{10} + 4 x^{6} + 4 x^{2} + 18$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62353	$x^{20} + 4 x^{14} + 8 x^{13} + 4 x^{10} + 4 x^{6} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62354	$x^{20} + 4 x^{14} + 8 x^{13} + 4 x^{10} + 4 x^{6} + 4 x^{2} + 18$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62355	$x^{20} + 8 x^{19} + 4 x^{14} + 8 x^{13} + 4 x^{10} + 4 x^{6} + 4 x^{2} + 2$	$C_2^8.(C_4\times F_5)$ (as 20T523)	$20480$	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62356	$x^{20} + 8 x^{19} + 4 x^{14} + 8 x^{13} + 4 x^{10} + 4 x^{6} + 4 x^{2} + 18$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62357	$x^{20} + 8 x^{17} + 4 x^{14} + 8 x^{13} + 4 x^{10} + 4 x^{6} + 4 x^{2} + 2$	$C_2^8.(C_4\times F_5)$ (as 20T523)	$20480$	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62358	$x^{20} + 8 x^{17} + 4 x^{14} + 8 x^{13} + 4 x^{10} + 4 x^{6} + 4 x^{2} + 18$	$C_2^8.(C_4\times F_5)$ (as 20T523)	$20480$	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62359	$x^{20} + 8 x^{19} + 8 x^{17} + 4 x^{14} + 8 x^{13} + 4 x^{10} + 4 x^{6} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62360	$x^{20} + 8 x^{19} + 8 x^{17} + 4 x^{14} + 8 x^{13} + 4 x^{10} + 4 x^{6} + 4 x^{2} + 18$	$C_2^8.(C_4\times F_5)$ (as 20T523)	$20480$	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62361	$x^{20} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 4 x^{10} + 4 x^{6} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62362	$x^{20} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 4 x^{10} + 4 x^{6} + 4 x^{2} + 18$	$C_2^8.(C_4\times F_5)$ (as 20T523)	$20480$	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62363	$x^{20} + 8 x^{19} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 4 x^{10} + 4 x^{6} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62364	$x^{20} + 8 x^{19} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 4 x^{10} + 4 x^{6} + 4 x^{2} + 18$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62365	$x^{20} + 8 x^{17} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 4 x^{10} + 4 x^{6} + 4 x^{2} + 2$	$C_2^8.(C_4\times F_5)$ (as 20T523)	$20480$	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62366	$x^{20} + 8 x^{17} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 4 x^{10} + 4 x^{6} + 4 x^{2} + 18$	$C_2^8.(C_4\times F_5)$ (as 20T523)	$20480$	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62367	$x^{20} + 8 x^{19} + 8 x^{17} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 4 x^{10} + 4 x^{6} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62368	$x^{20} + 8 x^{19} + 8 x^{17} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 4 x^{10} + 4 x^{6} + 4 x^{2} + 18$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62369	$x^{20} + 4 x^{14} + 4 x^{10} + 8 x^{9} + 4 x^{6} + 4 x^{2} + 2$	$C_2^9.C_2\wr F_5$ (as 20T909)	$327680$	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62370	$x^{20} + 8 x^{19} + 4 x^{14} + 4 x^{10} + 8 x^{9} + 4 x^{6} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62371	$x^{20} + 8 x^{17} + 4 x^{14} + 4 x^{10} + 8 x^{9} + 4 x^{6} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62372	$x^{20} + 8 x^{19} + 8 x^{17} + 4 x^{14} + 4 x^{10} + 8 x^{9} + 4 x^{6} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62373	$x^{20} + 8 x^{15} + 4 x^{14} + 4 x^{10} + 8 x^{9} + 4 x^{6} + 4 x^{2} + 2$	$C_2^9.C_2\wr F_5$ (as 20T909)	$327680$	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62374	$x^{20} + 8 x^{19} + 8 x^{15} + 4 x^{14} + 4 x^{10} + 8 x^{9} + 4 x^{6} + 4 x^{2} + 2$	$C_2^9.C_2\wr F_5$ (as 20T909)	$327680$	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62375	$x^{20} + 8 x^{17} + 8 x^{15} + 4 x^{14} + 4 x^{10} + 8 x^{9} + 4 x^{6} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62376	$x^{20} + 8 x^{19} + 8 x^{17} + 8 x^{15} + 4 x^{14} + 4 x^{10} + 8 x^{9} + 4 x^{6} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62377	$x^{20} + 4 x^{14} + 8 x^{13} + 4 x^{10} + 8 x^{9} + 4 x^{6} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62378	$x^{20} + 8 x^{19} + 4 x^{14} + 8 x^{13} + 4 x^{10} + 8 x^{9} + 4 x^{6} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62379	$x^{20} + 8 x^{17} + 4 x^{14} + 8 x^{13} + 4 x^{10} + 8 x^{9} + 4 x^{6} + 4 x^{2} + 2$	$C_2^9.C_2\wr F_5$ (as 20T909)	$327680$	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62380	$x^{20} + 8 x^{19} + 8 x^{17} + 4 x^{14} + 8 x^{13} + 4 x^{10} + 8 x^{9} + 4 x^{6} + 4 x^{2} + 2$	$C_2^9.C_2\wr F_5$ (as 20T909)	$327680$	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62381	$x^{20} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 4 x^{10} + 8 x^{9} + 4 x^{6} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62382	$x^{20} + 8 x^{19} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 4 x^{10} + 8 x^{9} + 4 x^{6} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62383	$x^{20} + 8 x^{17} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 4 x^{10} + 8 x^{9} + 4 x^{6} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62384	$x^{20} + 8 x^{19} + 8 x^{17} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 4 x^{10} + 8 x^{9} + 4 x^{6} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62385	$x^{20} + 4 x^{14} + 8 x^{11} + 4 x^{10} + 8 x^{9} + 4 x^{6} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$
2.1.20.59a1.62386	$x^{20} + 8 x^{19} + 4 x^{14} + 8 x^{11} + 4 x^{10} + 8 x^{9} + 4 x^{6} + 4 x^{2} + 2$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[40, 20, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 15, 35]$

  To download results, . Download displayed columns for results to