$p$-adic family 5.1.5.9a

• Label: 5.1.5.9a
• Base: 5.1.1.0a1.1
• Degree: \(5\)
• e: \(5\)
• f: \(1\)
• c: \(9\)

Defining polynomial

$x^{5} + 25b_{6} x + 5$

Invariants

Residue field characteristic:	$5$
Degree:	$5$
Base field:	$\Q_{5}$
Ramification index $e$:	$5$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$9$
Artin slopes:	$[\frac{9}{4}]$
Swan slopes:	$[\frac{5}{4}]$
Means:	$\langle1\rangle$
Rams:	$(\frac{5}{4})$
Field count:	$5$ (complete)
Ambiguity:	$1$
Mass:	$5$
Absolute Mass:	$5$

Diagrams

Varying

Indices of inseparability:	$[5,0]$
Associated inertia:	$[1]$
Jump Set:	undefined

Galois groups and Hidden Artin slopes

		Galois groups
		$F_5$ (as 5T3)
hidden slopes	$[\ ]_{4}$	5

Fields

Showing all 5

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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
5.1.5.9a1.1	5	$x^{5} + 5$	$F_5$ (as 5T3)	$20$	$1$	$[\frac{9}{4}]_{4}$	$[\frac{5}{4}]_{4}$	$[\ ]_{4}$	$[\ ]_{4}$	$[5, 0]$	$[1]$	$z + 1$	undefined
5.1.5.9a1.2	5	$x^{5} + 25 x + 5$	$F_5$ (as 5T3)	$20$	$1$	$[\frac{9}{4}]_{4}$	$[\frac{5}{4}]_{4}$	$[\ ]_{4}$	$[\ ]_{4}$	$[5, 0]$	$[1]$	$z + 1$	undefined
5.1.5.9a1.3	5	$x^{5} + 50 x + 5$	$F_5$ (as 5T3)	$20$	$1$	$[\frac{9}{4}]_{4}$	$[\frac{5}{4}]_{4}$	$[\ ]_{4}$	$[\ ]_{4}$	$[5, 0]$	$[1]$	$z + 1$	undefined
5.1.5.9a1.4	5	$x^{5} + 75 x + 5$	$F_5$ (as 5T3)	$20$	$1$	$[\frac{9}{4}]_{4}$	$[\frac{5}{4}]_{4}$	$[\ ]_{4}$	$[\ ]_{4}$	$[5, 0]$	$[1]$	$z + 1$	undefined
5.1.5.9a1.5	5	$x^{5} + 100 x + 5$	$F_5$ (as 5T3)	$20$	$1$	$[\frac{9}{4}]_{4}$	$[\frac{5}{4}]_{4}$	$[\ ]_{4}$	$[\ ]_{4}$	$[5, 0]$	$[1]$	$z + 1$	undefined

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