$p$-adic family 3.1.6.9a

• Label: 3.1.6.9a
• Base: 3.1.1.0a1.1
• Degree: \(6\)
• e: \(6\)
• f: \(1\)
• c: \(9\)

Defining polynomial

$x^{6} + 3 b_{5} x^{5} + 3 a_{4} x^{4} + 3 d_{0} + 9 c_{6}$

Invariants

Residue field characteristic:	$3$
Degree:	$6$
Base field:	$\Q_{3}$
Ramification index $e$:	$6$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$9$
Artin slopes:	$[2]$
Swan slopes:	$[1]$
Means:	$\langle\frac{2}{3}\rangle$
Rams:	$(2)$
Field count:	$16$ (complete)
Ambiguity:	$6$
Mass:	$6$
Absolute Mass:	$6$

Diagrams

Varying

Indices of inseparability:	$[4,0]$
Associated inertia:	$[1,1]$ (show 12), $[1,2]$ (show 4)
Jump Set:	undefined (show 8), $[1,3]$ (show 1), $[1,7]$ (show 2), $[1,8]$ (show 3), $[1,9]$ (show 2)

Galois groups and Hidden Artin slopes

Select desired size of Galois group.

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

		Galois groups of order 12
		$D_6$ (as 6T3)
hidden slopes	$[\ ]^{2}$	2

		Galois groups of order 18
		$C_3\times S_3$ (as 6T5)
hidden slopes	$[\frac{3}{2}]$	6

		Galois groups of order 36
		$S_3^2$ (as 6T9)
hidden slopes	$[\frac{3}{2}]^{2}$	2

Fields

Showing all 2

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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
3.1.6.9a1.2	6	$x^{6} + 6 x^{4} + 12$	$C_6$ (as 6T1)	$6$	$6$	$[2]_{2}$	$[1]_{2}$	$[\ ]$	$[\ ]$	$[4, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 1$	$[1, 9]$
3.1.6.9a1.3	6	$x^{6} + 6 x^{4} + 21$	$C_6$ (as 6T1)	$6$	$6$	$[2]_{2}$	$[1]_{2}$	$[\ ]$	$[\ ]$	$[4, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 1$	$[1, 9]$

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