$p$-adic family 7.3.3.6a

• Label: 7.3.3.6a
• Base: 7.1.1.0a1.1
• Degree: \(9\)
• e: \(3\)
• f: \(3\)
• c: \(6\)

Defining polynomial over unramified subextension

$x^{3} + 7d_{0}$

Invariants

Residue field characteristic:	$7$
Degree:	$9$
Base field:	$\Q_{7}$
Ramification index $e$:	$3$
Residue field degree $f$:	$3$
Discriminant exponent $c$:	$6$
Artin slopes:	$[\ ]$
Swan slopes:	$[\ ]$
Means:	$\langle\ \rangle$
Rams:	$(\ )$
Field count:	$3$ (complete)
Ambiguity:	$9$
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
		$C_9$ (as 9T1)	$C_3^2$ (as 9T2)	Total
hidden slopes	$[\ ]$	2	1	3

Fields

Showing all 3

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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
7.3.3.6a1.1	2	$( x^{3} + 6 x^{2} + 4 )^{3} + 7 x^{2}$	$C_9$ (as 9T1)	$9$	$9$	$[\ ]_{3}^{3}$	$[\ ]_{3}^{3}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^2 + 3 z + 3$	undefined
7.3.3.6a1.2	2	$( x^{3} + 6 x^{2} + 4 )^{3} + 7 x$	$C_9$ (as 9T1)	$9$	$9$	$[\ ]_{3}^{3}$	$[\ ]_{3}^{3}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^2 + 3 z + 3$	undefined
7.3.3.6a1.3	1	$( x^{3} + 6 x^{2} + 4 )^{3} + 7$	$C_3^2$ (as 9T2)	$9$	$9$	$[\ ]_{3}^{3}$	$[\ ]_{3}^{3}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^2 + 3 z + 3$	undefined

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