$p$-adic family 7.1.6.5a

• Label: 7.1.6.5a
• Base: 7.1.1.0a1.1
• Degree: \(6\)
• e: \(6\)
• f: \(1\)
• c: \(5\)

Defining polynomial

$x^{6} + 7d_{0}$

Invariants

Residue field characteristic:	$7$
Degree:	$6$
Base field:	$\Q_{7}$
Ramification index $e$:	$6$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$5$
Artin slopes:	$[\ ]$
Swan slopes:	$[\ ]$
Means:	$\langle\ \rangle$
Rams:	$(\ )$
Field count:	$6$ (complete)
Ambiguity:	$6$
Mass:	$1$
Absolute Mass:	$1$

Varying

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

Galois groups and Hidden Artin slopes

		Galois groups
		$C_6$ (as 6T1)
hidden slopes	$[\ ]$	6

Fields

Showing all 6

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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
7.1.6.5a1.1	6	$x^{6} + 7$	$C_6$ (as 6T1)	$6$	$6$	$[\ ]_{6}$	$[\ ]_{6}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^5 + 6 z^4 + z^3 + 6 z^2 + z + 6$	$[1]$
7.1.6.5a1.2	6	$x^{6} + 14$	$C_6$ (as 6T1)	$6$	$6$	$[\ ]_{6}$	$[\ ]_{6}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^5 + 6 z^4 + z^3 + 6 z^2 + z + 6$	undefined
7.1.6.5a1.3	6	$x^{6} + 21$	$C_6$ (as 6T1)	$6$	$6$	$[\ ]_{6}$	$[\ ]_{6}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^5 + 6 z^4 + z^3 + 6 z^2 + z + 6$	undefined
7.1.6.5a1.4	6	$x^{6} + 28$	$C_6$ (as 6T1)	$6$	$6$	$[\ ]_{6}$	$[\ ]_{6}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^5 + 6 z^4 + z^3 + 6 z^2 + z + 6$	undefined
7.1.6.5a1.5	6	$x^{6} + 35$	$C_6$ (as 6T1)	$6$	$6$	$[\ ]_{6}$	$[\ ]_{6}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^5 + 6 z^4 + z^3 + 6 z^2 + z + 6$	undefined
7.1.6.5a1.6	6	$x^{6} + 42$	$C_6$ (as 6T1)	$6$	$6$	$[\ ]_{6}$	$[\ ]_{6}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^5 + 6 z^4 + z^3 + 6 z^2 + z + 6$	undefined

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