$p$-adic family 3.3.3.12a

• Label: 3.3.3.12a
• Base: 3.1.1.0a1.1
• Degree: \(9\)
• e: \(3\)
• f: \(3\)
• c: \(12\)

Defining polynomial over unramified subextension

$x^{3} + 3 a_{2} x^{2} + 9 c_{3} + 3$

Invariants

Residue field characteristic:	$3$
Degree:	$9$
Base field:	$\Q_{3}$
Ramification index $e$:	$3$
Residue field degree $f$:	$3$
Discriminant exponent $c$:	$12$
Artin slopes:	$[2]$
Swan slopes:	$[1]$
Means:	$\langle\frac{2}{3}\rangle$
Rams:	$(1)$
Field count:	$20$ (complete)
Ambiguity:	$9$
Mass:	$26$
Absolute Mass:	$26/3$

Diagrams

Varying

Indices of inseparability:	$[2,0]$
Associated inertia:	$[1]$ (show 15), $[2]$ (show 5)
Jump Set:	undefined

Galois groups and Hidden Artin slopes

Select desired size of Galois group.

		Galois groups of order 9
		$C_9$ (as 9T1)	$C_3^2$ (as 9T2)	Total
hidden slopes	$[\ ]$	2	1	3

		Galois groups of order 18
		$C_3\times S_3$ (as 9T4)
hidden slopes	$[\ ]^{2}$	1

		Galois groups of order 27
		$C_9:C_3$ (as 9T6)	$\He_3$ (as 9T7)	Total
hidden slopes	$[2]$	2	1	3

		Galois groups of order 54
		$C_3^2:C_6$ (as 9T13)
hidden slopes	$[2]^{2}$	1

		Galois groups of order 81
		$C_3\wr C_3$ (as 9T17)
hidden slopes	$[2,2]$	9

		Galois groups of order 162
		$C_3^3:C_6$ (as 9T22)
hidden slopes	$[2,2]^{2}$	3

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
3.3.3.12a1.1	1	$( x^{3} + 2 x + 1 )^{3} + 3 ( x^{3} + 2 x + 1 )^{2} + 3$	$S_3\times C_3$ (as 9T4)	$18$	$3$	$[2]^{6}$	$[1]^{6}$	$[\ ]^{2}$	$[\ ]^{2}$	$[2, 0]$	$[2]$	$z^2 + 1$	undefined
3.3.3.12a4.1	1	$( x^{3} + 2 x + 1 )^{3} + 6 x ( x^{3} + 2 x + 1 )^{2} + 3$	$(C_3^3:C_3):C_2$ (as 9T22)	$162$	$1$	$[2, 2, 2]^{6}$	$[1,1,1]^{6}$	$[2,2]^{2}$	$[1,1]^{2}$	$[2, 0]$	$[2]$	$z^2 + 2 t$	undefined
3.3.3.12a5.1	1	$( x^{3} + 2 x + 1 )^{3} + 3 x^{2} ( x^{3} + 2 x + 1 )^{2} + 3$	$(C_3^3:C_3):C_2$ (as 9T22)	$162$	$1$	$[2, 2, 2]^{6}$	$[1,1,1]^{6}$	$[2,2]^{2}$	$[1,1]^{2}$	$[2, 0]$	$[2]$	$z^2 + t^2$	undefined
3.3.3.12a6.1	1	$( x^{3} + 2 x + 1 )^{3} + \left(3 x^{2} + 6 x\right) ( x^{3} + 2 x + 1 )^{2} + 3$	$(C_3^3:C_3):C_2$ (as 9T22)	$162$	$1$	$[2, 2, 2]^{6}$	$[1,1,1]^{6}$	$[2,2]^{2}$	$[1,1]^{2}$	$[2, 0]$	$[2]$	$z^2 + (t^2 + 2 t)$	undefined
3.3.3.12a8.1	1	$( x^{3} + 2 x + 1 )^{3} + \left(6 x^{2} + 6\right) ( x^{3} + 2 x + 1 )^{2} + 3$	$C_3^2 : S_3 $ (as 9T13)	$54$	$1$	$[2, 2]^{6}$	$[1,1]^{6}$	$[2]^{2}$	$[1]^{2}$	$[2, 0]$	$[2]$	$z^2 + (2 t^2 + 2)$	undefined

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