$p$-adic family 61.4.5.16a

• Label: 61.4.5.16a
• Base: 61.1.1.0a1.1
• Degree: \(20\)
• e: \(5\)
• f: \(4\)
• c: \(16\)

Defining polynomial over unramified subextension

$x^{5} + 61d_{0}$

Invariants

Residue field characteristic:	$61$
Degree:	$20$
Base field:	$\Q_{61}$
Ramification index $e$:	$5$
Residue field degree $f$:	$4$
Discriminant exponent $c$:	$16$
Artin slopes:	$[\ ]$
Swan slopes:	$[\ ]$
Means:	$\langle\ \rangle$
Rams:	$(\ )$
Field count:	$5$ (complete)
Ambiguity:	$20$
Mass:	$1$
Absolute Mass:	$1/4$

Varying

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

Galois groups and Hidden Artin slopes

		Galois groups
		$C_{20}$ (as 20T1)
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
61.4.5.16a1.1	5	$( x^{4} + 3 x^{2} + 40 x + 2 )^{5} + 61 x^{3}$	$C_{20}$ (as 20T1)	$20$	$20$	$[\ ]_{5}^{4}$	$[\ ]_{5}^{4}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^4 + 5 z^3 + 10 z^2 + 10 z + 5$	undefined
61.4.5.16a1.2	5	$( x^{4} + 3 x^{2} + 40 x + 2 )^{5} + 61 x^{2}$	$C_{20}$ (as 20T1)	$20$	$20$	$[\ ]_{5}^{4}$	$[\ ]_{5}^{4}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^4 + 5 z^3 + 10 z^2 + 10 z + 5$	undefined
61.4.5.16a1.3	5	$( x^{4} + 3 x^{2} + 40 x + 2 )^{5} + 61 x$	$C_{20}$ (as 20T1)	$20$	$20$	$[\ ]_{5}^{4}$	$[\ ]_{5}^{4}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^4 + 5 z^3 + 10 z^2 + 10 z + 5$	undefined
61.4.5.16a1.4	5	$( x^{4} + 3 x^{2} + 40 x + 2 )^{5} + 61$	$C_{20}$ (as 20T1)	$20$	$20$	$[\ ]_{5}^{4}$	$[\ ]_{5}^{4}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^4 + 5 z^3 + 10 z^2 + 10 z + 5$	undefined
61.4.5.16a1.5	5	$( x^{4} + 3 x^{2} + 40 x + 2 )^{5} + 3538 x^{2} + 1281 x + 3599$	$C_{20}$ (as 20T1)	$20$	$20$	$[\ ]_{5}^{4}$	$[\ ]_{5}^{4}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^4 + 5 z^3 + 10 z^2 + 10 z + 5$	undefined

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