$p$-adic family 5.1.20.20a

• Label: 5.1.20.20a
• Base: 5.1.1.0a1.1
• Degree: \(20\)
• e: \(20\)
• f: \(1\)
• c: \(20\)

Defining polynomial

$x^{20} + 5 a_{1} x + 5 d_{0}$

Invariants

Residue field characteristic:	$5$
Degree:	$20$
Base field:	$\Q_{5}$
Ramification index $e$:	$20$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$20$
Artin slopes:	$[\frac{17}{16}]$
Swan slopes:	$[\frac{1}{16}]$
Means:	$\langle\frac{1}{20}\rangle$
Rams:	$(\frac{1}{4})$
Field count:	$4$ (complete)
Ambiguity:	$4$
Mass:	$4$
Absolute Mass:	$4$

Diagrams

Varying

Indices of inseparability:	$[1,0]$
Associated inertia:	$[1,1]$
Jump Set:	undefined (show 3), $[1,6]$ (show 1)

Fields

Showing all 4

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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
5.1.20.20a1.1		$x^{20} + 5 x + 5$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[1, 0]$	$[1, 1]$	$z^{15} + 4 z^{10} + z^5 + 4,4 z + 4$	$[1, 6]$
5.1.20.20a1.2		$x^{20} + 5 x + 10$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[1, 0]$	$[1, 1]$	$z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$	undefined
5.1.20.20a1.3		$x^{20} + 5 x + 15$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[1, 0]$	$[1, 1]$	$z^{15} + 4 z^{10} + z^5 + 4,4 z + 3$	undefined
5.1.20.20a1.4		$x^{20} + 5 x + 20$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[1, 0]$	$[1, 1]$	$z^{15} + 4 z^{10} + z^5 + 4,4 z + 1$	undefined

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