$p$-adic family 109.2.3.4a

• Label: 109.2.3.4a
• Base: 109.1.1.0a1.1
• Degree: \(6\)
• e: \(3\)
• f: \(2\)
• c: \(4\)

Defining polynomial over unramified subextension

$x^{3} + 109d_{0}$

Invariants

Residue field characteristic:	$109$
Degree:	$6$
Base field:	$\Q_{109}$
Ramification index $e$:	$3$
Residue field degree $f$:	$2$
Discriminant exponent $c$:	$4$
Artin slopes:	$[\ ]$
Swan slopes:	$[\ ]$
Means:	$\langle\ \rangle$
Rams:	$(\ )$
Field count:	$3$ (complete)
Ambiguity:	$6$
Mass:	$1$
Absolute Mass:	$1/2$

Varying

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

Galois groups and Hidden Artin slopes

		Galois groups
		$C_6$ (as 6T1)
hidden slopes	$[\ ]$	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
109.2.3.4a1.1	3	$( x^{2} + 108 x + 6 )^{3} + 109 x$	$C_6$ (as 6T1)	$6$	$6$	$[\ ]_{3}^{2}$	$[\ ]_{3}^{2}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^2 + 3 z + 3$	undefined
109.2.3.4a1.2	3	$( x^{2} + 108 x + 6 )^{3} + 109$	$C_6$ (as 6T1)	$6$	$6$	$[\ ]_{3}^{2}$	$[\ ]_{3}^{2}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^2 + 3 z + 3$	undefined
109.2.3.4a1.3	3	$( x^{2} + 108 x + 6 )^{3} + 109 x + 11227$	$C_6$ (as 6T1)	$6$	$6$	$[\ ]_{3}^{2}$	$[\ ]_{3}^{2}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^2 + 3 z + 3$	undefined

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