$p$-adic family 2.1.20.30b

• Label: 2.1.20.30b
• Base: 2.1.1.0a1.1
• Degree: \(20\)
• e: \(20\)
• f: \(1\)
• c: \(30\)

Defining polynomial

$x^{20} + 2 b_{19} x^{19} + 2 b_{17} x^{17} + 2 b_{15} x^{15} + 2 b_{13} x^{13} + 2 a_{11} x^{11} + 2 c_{4} x^{4} + 2 a_{2} x^{2} + 4 c_{20} + 2$

Invariants

Residue field characteristic:	$2$
Degree:	$20$
Base field:	$\Q_{2}$
Ramification index $e$:	$20$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$30$
Artin slopes:	$[\frac{6}{5},2]$
Swan slopes:	$[\frac{1}{5},1]$
Means:	$\langle\frac{1}{10},\frac{11}{20}\rangle$
Rams:	$(1,9)$
Field count:	$40$ (complete)
Ambiguity:	$4$
Mass:	$16$
Absolute Mass:	$16$

Diagrams

Varying

Indices of inseparability:	$[11,2,0]$
Associated inertia:	$[4,1,1]$
Jump Set:	$[5,11,22]$ (show 2), $[5,11,33]$ (show 16), $[5,11,35]$ (show 12), $[5,11,37]$ (show 6), $[5,11,39]$ (show 2), $[5,11,40]$ (show 2)

Fields

Showing all 2

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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
2.1.20.30b1.7		$x^{20} + 2 x^{15} + 2 x^{11} + 2 x^{2} + 2$	not computed	$ $not computed$ $	$4$	not computed	not computed	not computed	not computed	$[11, 2, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 11, 22]$
2.1.20.30b1.28		$x^{20} + 2 x^{19} + 2 x^{15} + 2 x^{11} + 2 x^{4} + 2 x^{2} + 2$	$C_2\wr F_5$ (as 20T131)	$640$	$4$	not computed	not computed	not computed	not computed	$[11, 2, 0]$	$[4, 1, 1]$	$z^{16} + z^{12} + 1,z^2 + 1,z + 1$	$[5, 11, 22]$

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