$p$-adic family 2.1.20.24a

• Label: 2.1.20.24a
• Base: 2.1.1.0a1.1
• Degree: \(20\)
• e: \(20\)
• f: \(1\)
• c: \(24\)

Defining polynomial

$x^{20} + 2 b_{6} x^{6} + 2 a_{5} x^{5} + 2$

Invariants

Residue field characteristic:	$2$
Degree:	$20$
Base field:	$\Q_{2}$
Ramification index $e$:	$20$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$24$
Artin slopes:	$[\frac{4}{3},\frac{4}{3}]$
Swan slopes:	$[\frac{1}{3},\frac{1}{3}]$
Means:	$\langle\frac{1}{6},\frac{1}{4}\rangle$
Rams:	$(\frac{5}{3},\frac{5}{3})$
Field count:	$2$ (complete)
Ambiguity:	$1$
Mass:	$2$
Absolute Mass:	$2$

Diagrams

Varying

Indices of inseparability:	$[5,5,0]$
Associated inertia:	$[4,1]$
Jump Set:	$[5,10,25]$

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
2.1.20.24a1.1		$x^{20} + 2 x^{5} + 2$	$A_4:F_5$ (as 20T61)	$240$	$1$	not computed	not computed	not computed	not computed	$[5, 5, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 10, 25]$
2.1.20.24a1.2		$x^{20} + 2 x^{6} + 2 x^{5} + 2$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[5, 5, 0]$	$[4, 1]$	$z^{16} + z^{12} + 1,z + 1$	$[5, 10, 25]$

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