$p$-adic family 71.3.5.12a

• Label: 71.3.5.12a
• Base: 71.1.1.0a1.1
• Degree: \(15\)
• e: \(5\)
• f: \(3\)
• c: \(12\)

Defining polynomial over unramified subextension

$x^{5} + 71d_{0}$

Invariants

Residue field characteristic:	$71$
Degree:	$15$
Base field:	$\Q_{71}$
Ramification index $e$:	$5$
Residue field degree $f$:	$3$
Discriminant exponent $c$:	$12$
Artin slopes:	$[\ ]$
Swan slopes:	$[\ ]$
Means:	$\langle\ \rangle$
Rams:	$(\ )$
Field count:	$5$ (complete)
Ambiguity:	$15$
Mass:	$1$
Absolute Mass:	$1/3$

Varying

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

Galois groups and Hidden Artin slopes

		Galois groups
		$C_{15}$ (as 15T1)
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
71.3.5.12a1.1	5	$( x^{3} + 4 x + 64 )^{5} + 71 x^{2}$	$C_{15}$ (as 15T1)	$15$	$15$	$[\ ]_{5}^{3}$	$[\ ]_{5}^{3}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^4 + 5 z^3 + 10 z^2 + 10 z + 5$	undefined
71.3.5.12a1.2	5	$( x^{3} + 4 x + 64 )^{5} + 71 x$	$C_{15}$ (as 15T1)	$15$	$15$	$[\ ]_{5}^{3}$	$[\ ]_{5}^{3}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^4 + 5 z^3 + 10 z^2 + 10 z + 5$	undefined
71.3.5.12a1.3	5	$( x^{3} + 4 x + 64 )^{5} + 4757 x^{2} + 497 x$	$C_{15}$ (as 15T1)	$15$	$15$	$[\ ]_{5}^{3}$	$[\ ]_{5}^{3}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^4 + 5 z^3 + 10 z^2 + 10 z + 5$	undefined
71.3.5.12a1.4	5	$( x^{3} + 4 x + 64 )^{5} + 71$	$C_{15}$ (as 15T1)	$15$	$15$	$[\ ]_{5}^{3}$	$[\ ]_{5}^{3}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^4 + 5 z^3 + 10 z^2 + 10 z + 5$	undefined
71.3.5.12a1.5	5	$( x^{3} + 4 x + 64 )^{5} + 4757 x + 497$	$C_{15}$ (as 15T1)	$15$	$15$	$[\ ]_{5}^{3}$	$[\ ]_{5}^{3}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^4 + 5 z^3 + 10 z^2 + 10 z + 5$	undefined

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