$p$-adic family 5.1.20.26a

• Label: 5.1.20.26a
• Base: 5.1.1.0a1.1
• Degree: \(20\)
• e: \(20\)
• f: \(1\)
• c: \(26\)

Defining polynomial

$x^{20} + 5 b_{8} x^{8} + 5 a_{7} x^{7} + 5 d_{0}$

Invariants

Residue field characteristic:	$5$
Degree:	$20$
Base field:	$\Q_{5}$
Ramification index $e$:	$20$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$26$
Artin slopes:	$[\frac{23}{16}]$
Swan slopes:	$[\frac{7}{16}]$
Means:	$\langle\frac{7}{20}\rangle$
Rams:	$(\frac{7}{4})$
Field count:	$20$ (complete)
Ambiguity:	$4$
Mass:	$20$
Absolute Mass:	$20$

Diagrams

Varying

Indices of inseparability:	$[7,0]$
Associated inertia:	$[1,1]$
Jump Set:	undefined (show 15), $[1,12]$ (show 5)

Fields

Showing all 20

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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
5.1.20.26a1.1		$x^{20} + 5 x^{7} + 5$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[7, 0]$	$[1, 1]$	$z^{15} + 4 z^{10} + z^5 + 4,4 z + 3$	$[1, 12]$
5.1.20.26a1.2		$x^{20} + 5 x^{8} + 5 x^{7} + 5$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[7, 0]$	$[1, 1]$	$z^{15} + 4 z^{10} + z^5 + 4,4 z + 3$	$[1, 12]$
5.1.20.26a1.3		$x^{20} + 10 x^{8} + 5 x^{7} + 5$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[7, 0]$	$[1, 1]$	$z^{15} + 4 z^{10} + z^5 + 4,4 z + 3$	$[1, 12]$
5.1.20.26a1.4		$x^{20} + 15 x^{8} + 5 x^{7} + 5$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[7, 0]$	$[1, 1]$	$z^{15} + 4 z^{10} + z^5 + 4,4 z + 3$	$[1, 12]$
5.1.20.26a1.5		$x^{20} + 20 x^{8} + 5 x^{7} + 5$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[7, 0]$	$[1, 1]$	$z^{15} + 4 z^{10} + z^5 + 4,4 z + 3$	$[1, 12]$
5.1.20.26a1.6		$x^{20} + 5 x^{7} + 10$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[7, 0]$	$[1, 1]$	$z^{15} + 4 z^{10} + z^5 + 4,4 z + 4$	undefined
5.1.20.26a1.7		$x^{20} + 5 x^{8} + 5 x^{7} + 10$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[7, 0]$	$[1, 1]$	$z^{15} + 4 z^{10} + z^5 + 4,4 z + 4$	undefined
5.1.20.26a1.8		$x^{20} + 10 x^{8} + 5 x^{7} + 10$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[7, 0]$	$[1, 1]$	$z^{15} + 4 z^{10} + z^5 + 4,4 z + 4$	undefined
5.1.20.26a1.9		$x^{20} + 15 x^{8} + 5 x^{7} + 10$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[7, 0]$	$[1, 1]$	$z^{15} + 4 z^{10} + z^5 + 4,4 z + 4$	undefined
5.1.20.26a1.10		$x^{20} + 20 x^{8} + 5 x^{7} + 10$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[7, 0]$	$[1, 1]$	$z^{15} + 4 z^{10} + z^5 + 4,4 z + 4$	undefined
5.1.20.26a1.11		$x^{20} + 5 x^{7} + 15$	$C_5^4:C_{16}:C_4$ (as 20T605)	$40000$	$1$	not computed	not computed	not computed	not computed	$[7, 0]$	$[1, 1]$	$z^{15} + 4 z^{10} + z^5 + 4,4 z + 1$	undefined
5.1.20.26a1.12		$x^{20} + 5 x^{8} + 5 x^{7} + 15$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[7, 0]$	$[1, 1]$	$z^{15} + 4 z^{10} + z^5 + 4,4 z + 1$	undefined
5.1.20.26a1.13		$x^{20} + 10 x^{8} + 5 x^{7} + 15$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[7, 0]$	$[1, 1]$	$z^{15} + 4 z^{10} + z^5 + 4,4 z + 1$	undefined
5.1.20.26a1.14		$x^{20} + 15 x^{8} + 5 x^{7} + 15$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[7, 0]$	$[1, 1]$	$z^{15} + 4 z^{10} + z^5 + 4,4 z + 1$	undefined
5.1.20.26a1.15		$x^{20} + 20 x^{8} + 5 x^{7} + 15$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[7, 0]$	$[1, 1]$	$z^{15} + 4 z^{10} + z^5 + 4,4 z + 1$	undefined
5.1.20.26a1.16		$x^{20} + 5 x^{7} + 20$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[7, 0]$	$[1, 1]$	$z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$	undefined
5.1.20.26a1.17		$x^{20} + 5 x^{8} + 5 x^{7} + 20$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[7, 0]$	$[1, 1]$	$z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$	undefined
5.1.20.26a1.18		$x^{20} + 10 x^{8} + 5 x^{7} + 20$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[7, 0]$	$[1, 1]$	$z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$	undefined
5.1.20.26a1.19		$x^{20} + 15 x^{8} + 5 x^{7} + 20$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[7, 0]$	$[1, 1]$	$z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$	undefined
5.1.20.26a1.20		$x^{20} + 20 x^{8} + 5 x^{7} + 20$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[7, 0]$	$[1, 1]$	$z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$	undefined

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