$p$-adic family 2.2.10.38a

• Label: 2.2.10.38a
• Base: 2.1.1.0a1.1
• Degree: \(20\)
• e: \(10\)
• f: \(2\)
• c: \(38\)

Defining polynomial over unramified subextension

$x^{10} + 4 b_{19} x^{9} + 4 b_{17} x^{7} + 4 b_{15} x^{5} + 4 b_{13} x^{3} + 4 b_{11} x + 8 c_{20} + 2$

Invariants

Residue field characteristic:	$2$
Degree:	$20$
Base field:	$\Q_{2}$
Ramification index $e$:	$10$
Residue field degree $f$:	$2$
Discriminant exponent $c$:	$38$
Artin slopes:	$[3]$
Swan slopes:	$[2]$
Means:	$\langle1\rangle$
Rams:	$(10)$
Field count:	$1056$ (complete)
Ambiguity:	$4$
Mass:	$1024$
Absolute Mass:	$512$

Diagrams

Varying

Indices of inseparability:	$[10,0]$
Associated inertia:	$[2,1]$
Jump Set:	$[5,15]$

Fields

Showing 1-50 of

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Label	Packet size	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
2.2.10.38a1.1		$( x^{2} + x + 1 )^{10} + 2$	not computed	$ $not computed$ $	$4$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.2		$( x^{2} + x + 1 )^{10} + 8 x + 2$	not computed	$ $not computed$ $	$4$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.3		$( x^{2} + x + 1 )^{10} + 4 x ( x^{2} + x + 1 )^{9} + 2$	$C_2\wr F_5$ (as 20T134)	$640$	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.4		$( x^{2} + x + 1 )^{10} + 4 x ( x^{2} + x + 1 )^{9} + 8 x + 2$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.5		$( x^{2} + x + 1 )^{10} + 4 ( x^{2} + x + 1 )^{9} + 2$	not computed	$ $not computed$ $	$4$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.6		$( x^{2} + x + 1 )^{10} + 4 ( x^{2} + x + 1 )^{9} + 8 x + 2$	$C_2\wr F_5$ (as 20T135)	$640$	$4$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.7		$( x^{2} + x + 1 )^{10} + 4 x ( x^{2} + x + 1 )^{7} + 2$	$C_2\wr F_5$ (as 20T134)	$640$	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.8		$( x^{2} + x + 1 )^{10} + 4 x ( x^{2} + x + 1 )^{7} + 8 x + 2$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.9		$( x^{2} + x + 1 )^{10} + 4 x ( x^{2} + x + 1 )^{9} + 4 x ( x^{2} + x + 1 )^{7} + 2$	$C_2^9.F_5$ (as 20T429)	$10240$	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.10		$( x^{2} + x + 1 )^{10} + 4 x ( x^{2} + x + 1 )^{9} + 4 x ( x^{2} + x + 1 )^{7} + 8 x + 2$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.11		$( x^{2} + x + 1 )^{10} + 4 ( x^{2} + x + 1 )^{9} + 4 x ( x^{2} + x + 1 )^{7} + 2$	$C_2^9.F_5$ (as 20T429)	$10240$	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.12		$( x^{2} + x + 1 )^{10} + 4 ( x^{2} + x + 1 )^{9} + 4 x ( x^{2} + x + 1 )^{7} + 8 x + 2$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.13		$( x^{2} + x + 1 )^{10} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{9} + 4 x ( x^{2} + x + 1 )^{7} + 2$	$C_2\wr F_5$ (as 20T134)	$640$	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.14		$( x^{2} + x + 1 )^{10} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{9} + 4 x ( x^{2} + x + 1 )^{7} + 8 x + 2$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.15		$( x^{2} + x + 1 )^{10} + 4 ( x^{2} + x + 1 )^{7} + 2$	$C_2\wr F_5$ (as 20T140)	$640$	$4$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.16		$( x^{2} + x + 1 )^{10} + 4 ( x^{2} + x + 1 )^{7} + 8 x + 2$	$C_2\wr F_5$ (as 20T135)	$640$	$4$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.17		$( x^{2} + x + 1 )^{10} + 4 x ( x^{2} + x + 1 )^{9} + 4 ( x^{2} + x + 1 )^{7} + 2$	$C_2^9.F_5$ (as 20T429)	$10240$	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.18		$( x^{2} + x + 1 )^{10} + 4 x ( x^{2} + x + 1 )^{9} + 4 ( x^{2} + x + 1 )^{7} + 8 x + 2$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.19		$( x^{2} + x + 1 )^{10} + 4 ( x^{2} + x + 1 )^{9} + 4 ( x^{2} + x + 1 )^{7} + 2$	not computed	$ $not computed$ $	$4$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.20		$( x^{2} + x + 1 )^{10} + 4 ( x^{2} + x + 1 )^{9} + 4 ( x^{2} + x + 1 )^{7} + 8 x + 2$	$C_2\wr F_5$ (as 20T135)	$640$	$4$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.21		$( x^{2} + x + 1 )^{10} + 4 x ( x^{2} + x + 1 )^{5} + 2$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.22		$( x^{2} + x + 1 )^{10} + 4 x ( x^{2} + x + 1 )^{5} + 8 x + 2$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.23		$( x^{2} + x + 1 )^{10} + 4 x ( x^{2} + x + 1 )^{9} + 4 x ( x^{2} + x + 1 )^{5} + 2$	$C_2^{10}.F_5$ (as 20T513)	$20480$	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.24		$( x^{2} + x + 1 )^{10} + 4 x ( x^{2} + x + 1 )^{9} + 4 x ( x^{2} + x + 1 )^{5} + 8 x + 2$	$C_2^{10}.F_5$ (as 20T513)	$20480$	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.25		$( x^{2} + x + 1 )^{10} + 4 ( x^{2} + x + 1 )^{9} + 4 x ( x^{2} + x + 1 )^{5} + 2$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.26		$( x^{2} + x + 1 )^{10} + 4 ( x^{2} + x + 1 )^{9} + 4 x ( x^{2} + x + 1 )^{5} + 8 x + 2$	$C_2^6:F_5$ (as 20T191)	$1280$	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.27		$( x^{2} + x + 1 )^{10} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{9} + 4 x ( x^{2} + x + 1 )^{5} + 2$	$C_2^{10}.F_5$ (as 20T513)	$20480$	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.28		$( x^{2} + x + 1 )^{10} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{9} + 4 x ( x^{2} + x + 1 )^{5} + 8 x + 2$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.29		$( x^{2} + x + 1 )^{10} + 4 x ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{5} + 2$	$C_2^6:F_5$ (as 20T192)	$1280$	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.30		$( x^{2} + x + 1 )^{10} + 4 x ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{5} + 8 x + 2$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.31		$( x^{2} + x + 1 )^{10} + 4 x ( x^{2} + x + 1 )^{9} + 4 x ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{5} + 2$	$C_2^6:F_5$ (as 20T192)	$1280$	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.32		$( x^{2} + x + 1 )^{10} + 4 x ( x^{2} + x + 1 )^{9} + 4 x ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{5} + 8 x + 2$	$C_2^6:F_5$ (as 20T192)	$1280$	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.33		$( x^{2} + x + 1 )^{10} + 4 ( x^{2} + x + 1 )^{9} + 4 x ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{5} + 2$	$C_2^{10}.F_5$ (as 20T513)	$20480$	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.34		$( x^{2} + x + 1 )^{10} + 4 ( x^{2} + x + 1 )^{9} + 4 x ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{5} + 8 x + 2$	$C_2^{10}.F_5$ (as 20T513)	$20480$	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.35		$( x^{2} + x + 1 )^{10} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{9} + 4 x ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{5} + 2$	$C_2^{10}.F_5$ (as 20T513)	$20480$	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.36		$( x^{2} + x + 1 )^{10} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{9} + 4 x ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{5} + 8 x + 2$	$C_2^{10}.F_5$ (as 20T513)	$20480$	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.37		$( x^{2} + x + 1 )^{10} + 4 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{5} + 2$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.38		$( x^{2} + x + 1 )^{10} + 4 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{5} + 8 x + 2$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.39		$( x^{2} + x + 1 )^{10} + 4 x ( x^{2} + x + 1 )^{9} + 4 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{5} + 2$	$C_2^6:F_5$ (as 20T192)	$1280$	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.40		$( x^{2} + x + 1 )^{10} + 4 x ( x^{2} + x + 1 )^{9} + 4 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{5} + 8 x + 2$	$C_2^6:F_5$ (as 20T192)	$1280$	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.41		$( x^{2} + x + 1 )^{10} + 4 ( x^{2} + x + 1 )^{9} + 4 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{5} + 2$	$C_2^6:F_5$ (as 20T191)	$1280$	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.42		$( x^{2} + x + 1 )^{10} + 4 ( x^{2} + x + 1 )^{9} + 4 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{5} + 8 x + 2$	$C_2^6:F_5$ (as 20T191)	$1280$	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.43		$( x^{2} + x + 1 )^{10} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{9} + 4 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{5} + 2$	$C_2^6:F_5$ (as 20T192)	$1280$	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.44		$( x^{2} + x + 1 )^{10} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{9} + 4 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{5} + 8 x + 2$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.45		$( x^{2} + x + 1 )^{10} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{5} + 2$	$C_2^6:F_5$ (as 20T192)	$1280$	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.46		$( x^{2} + x + 1 )^{10} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{5} + 8 x + 2$	$C_2^6:F_5$ (as 20T192)	$1280$	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.47		$( x^{2} + x + 1 )^{10} + 4 x ( x^{2} + x + 1 )^{9} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{5} + 2$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.48		$( x^{2} + x + 1 )^{10} + 4 x ( x^{2} + x + 1 )^{9} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{5} + 8 x + 2$	$C_2^{10}.F_5$ (as 20T513)	$20480$	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.49		$( x^{2} + x + 1 )^{10} + 4 ( x^{2} + x + 1 )^{9} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{5} + 2$	$C_2^{10}.F_5$ (as 20T513)	$20480$	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$
2.2.10.38a1.50		$( x^{2} + x + 1 )^{10} + 4 ( x^{2} + x + 1 )^{9} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{5} + 8 x + 2$	$C_2^{10}.F_5$ (as 20T513)	$20480$	$2$	not computed	not computed	not computed	not computed	$[10, 0]$	$[2, 1]$	$z^8 + z^6 + 1,z + 1$	$[5, 15]$

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