$p$-adic family 41.1.5.4a

• Label: 41.1.5.4a
• Base: 41.1.1.0a1.1
• Degree: \(5\)
• e: \(5\)
• f: \(1\)
• c: \(4\)

Defining polynomial

$x^{5} + 41d_{0}$

Invariants

Residue field characteristic:	$41$
Degree:	$5$
Base field:	$\Q_{41}$
Ramification index $e$:	$5$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$4$
Artin slopes:	$[\ ]$
Swan slopes:	$[\ ]$
Means:	$\langle\ \rangle$
Rams:	$(\ )$
Field count:	$5$ (complete)
Ambiguity:	$5$
Mass:	$1$
Absolute Mass:	$1$

Varying

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

Galois groups and Hidden Artin slopes

		Galois groups
		$C_5$ (as 5T1)
hidden slopes	$[\ ]$	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
41.1.5.4a1.1	5	$x^{5} + 41$	$C_5$ (as 5T1)	$5$	$5$	$[\ ]_{5}$	$[\ ]_{5}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^4 + 5 z^3 + 10 z^2 + 10 z + 5$	undefined
41.1.5.4a1.2	5	$x^{5} + 246$	$C_5$ (as 5T1)	$5$	$5$	$[\ ]_{5}$	$[\ ]_{5}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^4 + 5 z^3 + 10 z^2 + 10 z + 5$	undefined
41.1.5.4a1.3	5	$x^{5} + 451$	$C_5$ (as 5T1)	$5$	$5$	$[\ ]_{5}$	$[\ ]_{5}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^4 + 5 z^3 + 10 z^2 + 10 z + 5$	undefined
41.1.5.4a1.4	5	$x^{5} + 1025$	$C_5$ (as 5T1)	$5$	$5$	$[\ ]_{5}$	$[\ ]_{5}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^4 + 5 z^3 + 10 z^2 + 10 z + 5$	undefined
41.1.5.4a1.5	5	$x^{5} + 1476$	$C_5$ (as 5T1)	$5$	$5$	$[\ ]_{5}$	$[\ ]_{5}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^4 + 5 z^3 + 10 z^2 + 10 z + 5$	undefined

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