$p$-adic family 3.2.6.22a

• Label: 3.2.6.22a
• Base: 3.1.1.0a1.1
• Degree: \(12\)
• e: \(6\)
• f: \(2\)
• c: \(22\)

Defining polynomial over unramified subextension

$x^{6} + 9 c_{9} x^{3} + 9 b_{8} x^{2} + 9 b_{7} x + 3 d_{0}$

Invariants

Residue field characteristic:	$3$
Degree:	$12$
Base field:	$\Q_{3}$
Ramification index $e$:	$6$
Residue field degree $f$:	$2$
Discriminant exponent $c$:	$22$
Artin slopes:	$[\frac{5}{2}]$
Swan slopes:	$[\frac{3}{2}]$
Means:	$\langle1\rangle$
Rams:	$(3)$
Field count:	$96$ (complete)
Ambiguity:	$12$
Mass:	$81$
Absolute Mass:	$81/2$

Diagrams

Varying

Indices of inseparability:	$[6,0]$
Associated inertia:	$[1,1]$ (show 72), $[1,2]$ (show 24)
Jump Set:	undefined (show 24), $[1,7]$ (show 72)

Galois groups and Hidden Artin slopes

Select desired size of Galois group.

		Galois groups of order 12
		$D_6$ (as 12T3)
hidden slopes	$[\ ]$	3

		Galois groups of order 24
		$C_4\times S_3$ (as 12T11)
hidden slopes	$[\ ]^{2}$	3

		Galois groups of order 36
		$S_3^2$ (as 12T16)	$C_6\times S_3$ (as 12T18)	Total
hidden slopes	$[2]$	3	9	12
		$[\frac{3}{2}]$	3		3
	$[\ ]^{3}$			3	3
		Total	6	12	18

		Galois groups of order 72
		$C_6.D_6$ (as 12T39)
hidden slopes	$[\frac{3}{2}]^{2}$	3

		Galois groups of order 108
		$C_3\times S_3^2$ (as 12T70)	$C_3:S_3^2$ (as 12T71)	Total
hidden slopes	$[2,2]$	9		9
		$[2]^{3}$	3		3
	$[\frac{3}{2},2]$		9	3	12
		$[\frac{3}{2}]^{3}$	3		3
	Total		24	3	27

		Galois groups of order 216
		$C_3^2:C_4\times S_3$ (as 12T119)
hidden slopes	$[2,2]^{2}$	6

		Galois groups of order 324
		$C_3\wr C_2^2$ (as 12T130)
hidden slopes	$[\frac{3}{2},2,2]$	18
		$[\frac{3}{2},2]^{3}$	6
	Total		24

		Galois groups of order 648
		$C_3^3:(C_4\times S_3)$ (as 12T170)
hidden slopes	$[\frac{3}{2},2,2]^{2}$	12

Fields

Showing all 24

  Download displayed columns for results to

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
3.2.6.22a2.1	12	$( x^{2} + 2 x + 2 )^{6} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 ) + 3 x$	$C_3^3:(C_4\times S_3)$ (as 12T170)	$648$	$1$	$[\frac{3}{2}, 2, 2, \frac{5}{2}]_{2}^{4}$	$[\frac{1}{2},1,1,\frac{3}{2}]_{2}^{4}$	$[\frac{3}{2},2,2]^{2}$	$[\frac{1}{2},1,1]^{2}$	$[6, 0]$	$[1, 2]$	$z^3 + 2,2 z^2 + (2 t + 1)$	undefined
3.2.6.22a2.2	12	$( x^{2} + 2 x + 2 )^{6} + 9 x ( x^{2} + 2 x + 2 )^{2} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 ) + 3 x$	$C_3^3:(C_4\times S_3)$ (as 12T170)	$648$	$1$	$[\frac{3}{2}, 2, 2, \frac{5}{2}]_{2}^{4}$	$[\frac{1}{2},1,1,\frac{3}{2}]_{2}^{4}$	$[\frac{3}{2},2,2]^{2}$	$[\frac{1}{2},1,1]^{2}$	$[6, 0]$	$[1, 2]$	$z^3 + 2,2 z^2 + (2 t + 1)$	undefined
3.2.6.22a2.3	6	$( x^{2} + 2 x + 2 )^{6} + 18 x ( x^{2} + 2 x + 2 )^{2} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 ) + 3 x$	$C_3^2:C_4\times S_3$ (as 12T119)	$216$	$1$	$[2, 2, \frac{5}{2}]_{2}^{4}$	$[1,1,\frac{3}{2}]_{2}^{4}$	$[2,2]^{2}$	$[1,1]^{2}$	$[6, 0]$	$[1, 2]$	$z^3 + 2,2 z^2 + (2 t + 1)$	undefined
3.2.6.22a2.4	6	$( x^{2} + 2 x + 2 )^{6} + 9 ( x^{2} + 2 x + 2 )^{2} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 ) + 3 x$	$C_3^2:C_4\times S_3$ (as 12T119)	$216$	$1$	$[2, 2, \frac{5}{2}]_{2}^{4}$	$[1,1,\frac{3}{2}]_{2}^{4}$	$[2,2]^{2}$	$[1,1]^{2}$	$[6, 0]$	$[1, 2]$	$z^3 + 2,2 z^2 + (2 t + 1)$	undefined
3.2.6.22a2.5	12	$( x^{2} + 2 x + 2 )^{6} + \left(9 x + 9\right) ( x^{2} + 2 x + 2 )^{2} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 ) + 3 x$	$C_3^3:(C_4\times S_3)$ (as 12T170)	$648$	$1$	$[\frac{3}{2}, 2, 2, \frac{5}{2}]_{2}^{4}$	$[\frac{1}{2},1,1,\frac{3}{2}]_{2}^{4}$	$[\frac{3}{2},2,2]^{2}$	$[\frac{1}{2},1,1]^{2}$	$[6, 0]$	$[1, 2]$	$z^3 + 2,2 z^2 + (2 t + 1)$	undefined
3.2.6.22a2.6	12	$( x^{2} + 2 x + 2 )^{6} + \left(18 x + 9\right) ( x^{2} + 2 x + 2 )^{2} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 ) + 3 x$	$C_3^3:(C_4\times S_3)$ (as 12T170)	$648$	$1$	$[\frac{3}{2}, 2, 2, \frac{5}{2}]_{2}^{4}$	$[\frac{1}{2},1,1,\frac{3}{2}]_{2}^{4}$	$[\frac{3}{2},2,2]^{2}$	$[\frac{1}{2},1,1]^{2}$	$[6, 0]$	$[1, 2]$	$z^3 + 2,2 z^2 + (2 t + 1)$	undefined
3.2.6.22a2.7	12	$( x^{2} + 2 x + 2 )^{6} + 18 ( x^{2} + 2 x + 2 )^{2} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 ) + 3 x$	$C_3^3:(C_4\times S_3)$ (as 12T170)	$648$	$1$	$[\frac{3}{2}, 2, 2, \frac{5}{2}]_{2}^{4}$	$[\frac{1}{2},1,1,\frac{3}{2}]_{2}^{4}$	$[\frac{3}{2},2,2]^{2}$	$[\frac{1}{2},1,1]^{2}$	$[6, 0]$	$[1, 2]$	$z^3 + 2,2 z^2 + (2 t + 1)$	undefined
3.2.6.22a2.8	6	$( x^{2} + 2 x + 2 )^{6} + \left(9 x + 18\right) ( x^{2} + 2 x + 2 )^{2} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 ) + 3 x$	$C_3^2:C_4\times S_3$ (as 12T119)	$216$	$1$	$[2, 2, \frac{5}{2}]_{2}^{4}$	$[1,1,\frac{3}{2}]_{2}^{4}$	$[2,2]^{2}$	$[1,1]^{2}$	$[6, 0]$	$[1, 2]$	$z^3 + 2,2 z^2 + (2 t + 1)$	undefined
3.2.6.22a2.9	12	$( x^{2} + 2 x + 2 )^{6} + \left(18 x + 18\right) ( x^{2} + 2 x + 2 )^{2} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 ) + 3 x$	$C_3^3:(C_4\times S_3)$ (as 12T170)	$648$	$1$	$[\frac{3}{2}, 2, 2, \frac{5}{2}]_{2}^{4}$	$[\frac{1}{2},1,1,\frac{3}{2}]_{2}^{4}$	$[\frac{3}{2},2,2]^{2}$	$[\frac{1}{2},1,1]^{2}$	$[6, 0]$	$[1, 2]$	$z^3 + 2,2 z^2 + (2 t + 1)$	undefined
3.2.6.22a2.10	3	$( x^{2} + 2 x + 2 )^{6} + \left(15 x + 6\right) ( x^{2} + 2 x + 2 ) + 3 x$	$S_3 \times C_4$ (as 12T11)	$24$	$4$	$[\frac{5}{2}]_{2}^{4}$	$[\frac{3}{2}]_{2}^{4}$	$[\ ]^{2}$	$[\ ]^{2}$	$[6, 0]$	$[1, 2]$	$z^3 + 2,2 z^2 + (2 t + 1)$	undefined
3.2.6.22a2.11	3	$( x^{2} + 2 x + 2 )^{6} + 9 x ( x^{2} + 2 x + 2 )^{2} + \left(15 x + 6\right) ( x^{2} + 2 x + 2 ) + 3 x$	$C_6.D_6$ (as 12T39)	$72$	$2$	$[\frac{3}{2}, \frac{5}{2}]_{2}^{4}$	$[\frac{1}{2},\frac{3}{2}]_{2}^{4}$	$[\frac{3}{2}]^{2}$	$[\frac{1}{2}]^{2}$	$[6, 0]$	$[1, 2]$	$z^3 + 2,2 z^2 + (2 t + 1)$	undefined
3.2.6.22a2.12	3	$( x^{2} + 2 x + 2 )^{6} + 18 x ( x^{2} + 2 x + 2 )^{2} + \left(15 x + 6\right) ( x^{2} + 2 x + 2 ) + 3 x$	$C_6.D_6$ (as 12T39)	$72$	$2$	$[\frac{3}{2}, \frac{5}{2}]_{2}^{4}$	$[\frac{1}{2},\frac{3}{2}]_{2}^{4}$	$[\frac{3}{2}]^{2}$	$[\frac{1}{2}]^{2}$	$[6, 0]$	$[1, 2]$	$z^3 + 2,2 z^2 + (2 t + 1)$	undefined
3.2.6.22a2.13	3	$( x^{2} + 2 x + 2 )^{6} + 9 ( x^{2} + 2 x + 2 )^{2} + \left(15 x + 6\right) ( x^{2} + 2 x + 2 ) + 3 x$	$C_6.D_6$ (as 12T39)	$72$	$2$	$[\frac{3}{2}, \frac{5}{2}]_{2}^{4}$	$[\frac{1}{2},\frac{3}{2}]_{2}^{4}$	$[\frac{3}{2}]^{2}$	$[\frac{1}{2}]^{2}$	$[6, 0]$	$[1, 2]$	$z^3 + 2,2 z^2 + (2 t + 1)$	undefined
3.2.6.22a2.14	3	$( x^{2} + 2 x + 2 )^{6} + \left(9 x + 9\right) ( x^{2} + 2 x + 2 )^{2} + \left(15 x + 6\right) ( x^{2} + 2 x + 2 ) + 3 x$	$S_3 \times C_4$ (as 12T11)	$24$	$4$	$[\frac{5}{2}]_{2}^{4}$	$[\frac{3}{2}]_{2}^{4}$	$[\ ]^{2}$	$[\ ]^{2}$	$[6, 0]$	$[1, 2]$	$z^3 + 2,2 z^2 + (2 t + 1)$	undefined
3.2.6.22a2.15	3	$( x^{2} + 2 x + 2 )^{6} + \left(18 x + 18\right) ( x^{2} + 2 x + 2 )^{2} + \left(15 x + 6\right) ( x^{2} + 2 x + 2 ) + 3 x$	$S_3 \times C_4$ (as 12T11)	$24$	$4$	$[\frac{5}{2}]_{2}^{4}$	$[\frac{3}{2}]_{2}^{4}$	$[\ ]^{2}$	$[\ ]^{2}$	$[6, 0]$	$[1, 2]$	$z^3 + 2,2 z^2 + (2 t + 1)$	undefined
3.2.6.22a2.16	6	$( x^{2} + 2 x + 2 )^{6} + \left(6 x + 15\right) ( x^{2} + 2 x + 2 ) + 3 x$	$C_3^2:C_4\times S_3$ (as 12T119)	$216$	$1$	$[2, 2, \frac{5}{2}]_{2}^{4}$	$[1,1,\frac{3}{2}]_{2}^{4}$	$[2,2]^{2}$	$[1,1]^{2}$	$[6, 0]$	$[1, 2]$	$z^3 + 2,2 z^2 + (2 t + 1)$	undefined
3.2.6.22a2.17	12	$( x^{2} + 2 x + 2 )^{6} + 9 x ( x^{2} + 2 x + 2 )^{2} + \left(6 x + 15\right) ( x^{2} + 2 x + 2 ) + 3 x$	$C_3^3:(C_4\times S_3)$ (as 12T170)	$648$	$1$	$[\frac{3}{2}, 2, 2, \frac{5}{2}]_{2}^{4}$	$[\frac{1}{2},1,1,\frac{3}{2}]_{2}^{4}$	$[\frac{3}{2},2,2]^{2}$	$[\frac{1}{2},1,1]^{2}$	$[6, 0]$	$[1, 2]$	$z^3 + 2,2 z^2 + (2 t + 1)$	undefined
3.2.6.22a2.18	12	$( x^{2} + 2 x + 2 )^{6} + 18 x ( x^{2} + 2 x + 2 )^{2} + \left(6 x + 15\right) ( x^{2} + 2 x + 2 ) + 3 x$	$C_3^3:(C_4\times S_3)$ (as 12T170)	$648$	$1$	$[\frac{3}{2}, 2, 2, \frac{5}{2}]_{2}^{4}$	$[\frac{1}{2},1,1,\frac{3}{2}]_{2}^{4}$	$[\frac{3}{2},2,2]^{2}$	$[\frac{1}{2},1,1]^{2}$	$[6, 0]$	$[1, 2]$	$z^3 + 2,2 z^2 + (2 t + 1)$	undefined
3.2.6.22a2.19	12	$( x^{2} + 2 x + 2 )^{6} + 9 ( x^{2} + 2 x + 2 )^{2} + \left(6 x + 15\right) ( x^{2} + 2 x + 2 ) + 3 x$	$C_3^3:(C_4\times S_3)$ (as 12T170)	$648$	$1$	$[\frac{3}{2}, 2, 2, \frac{5}{2}]_{2}^{4}$	$[\frac{1}{2},1,1,\frac{3}{2}]_{2}^{4}$	$[\frac{3}{2},2,2]^{2}$	$[\frac{1}{2},1,1]^{2}$	$[6, 0]$	$[1, 2]$	$z^3 + 2,2 z^2 + (2 t + 1)$	undefined
3.2.6.22a2.20	6	$( x^{2} + 2 x + 2 )^{6} + \left(9 x + 9\right) ( x^{2} + 2 x + 2 )^{2} + \left(6 x + 15\right) ( x^{2} + 2 x + 2 ) + 3 x$	$C_3^2:C_4\times S_3$ (as 12T119)	$216$	$1$	$[2, 2, \frac{5}{2}]_{2}^{4}$	$[1,1,\frac{3}{2}]_{2}^{4}$	$[2,2]^{2}$	$[1,1]^{2}$	$[6, 0]$	$[1, 2]$	$z^3 + 2,2 z^2 + (2 t + 1)$	undefined
3.2.6.22a2.21	12	$( x^{2} + 2 x + 2 )^{6} + \left(18 x + 9\right) ( x^{2} + 2 x + 2 )^{2} + \left(6 x + 15\right) ( x^{2} + 2 x + 2 ) + 3 x$	$C_3^3:(C_4\times S_3)$ (as 12T170)	$648$	$1$	$[\frac{3}{2}, 2, 2, \frac{5}{2}]_{2}^{4}$	$[\frac{1}{2},1,1,\frac{3}{2}]_{2}^{4}$	$[\frac{3}{2},2,2]^{2}$	$[\frac{1}{2},1,1]^{2}$	$[6, 0]$	$[1, 2]$	$z^3 + 2,2 z^2 + (2 t + 1)$	undefined
3.2.6.22a2.22	12	$( x^{2} + 2 x + 2 )^{6} + 18 ( x^{2} + 2 x + 2 )^{2} + \left(6 x + 15\right) ( x^{2} + 2 x + 2 ) + 3 x$	$C_3^3:(C_4\times S_3)$ (as 12T170)	$648$	$1$	$[\frac{3}{2}, 2, 2, \frac{5}{2}]_{2}^{4}$	$[\frac{1}{2},1,1,\frac{3}{2}]_{2}^{4}$	$[\frac{3}{2},2,2]^{2}$	$[\frac{1}{2},1,1]^{2}$	$[6, 0]$	$[1, 2]$	$z^3 + 2,2 z^2 + (2 t + 1)$	undefined
3.2.6.22a2.23	12	$( x^{2} + 2 x + 2 )^{6} + \left(9 x + 18\right) ( x^{2} + 2 x + 2 )^{2} + \left(6 x + 15\right) ( x^{2} + 2 x + 2 ) + 3 x$	$C_3^3:(C_4\times S_3)$ (as 12T170)	$648$	$1$	$[\frac{3}{2}, 2, 2, \frac{5}{2}]_{2}^{4}$	$[\frac{1}{2},1,1,\frac{3}{2}]_{2}^{4}$	$[\frac{3}{2},2,2]^{2}$	$[\frac{1}{2},1,1]^{2}$	$[6, 0]$	$[1, 2]$	$z^3 + 2,2 z^2 + (2 t + 1)$	undefined
3.2.6.22a2.24	6	$( x^{2} + 2 x + 2 )^{6} + \left(18 x + 18\right) ( x^{2} + 2 x + 2 )^{2} + \left(6 x + 15\right) ( x^{2} + 2 x + 2 ) + 3 x$	$C_3^2:C_4\times S_3$ (as 12T119)	$216$	$1$	$[2, 2, \frac{5}{2}]_{2}^{4}$	$[1,1,\frac{3}{2}]_{2}^{4}$	$[2,2]^{2}$	$[1,1]^{2}$	$[6, 0]$	$[1, 2]$	$z^3 + 2,2 z^2 + (2 t + 1)$	undefined

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