$p$-adic family 3.1.12.21a

• Label: 3.1.12.21a
• Base: 3.1.1.0a1.1
• Degree: \(12\)
• e: \(12\)
• f: \(1\)
• c: \(21\)

Defining polynomial

$x^{12} + 3 b_{11} x^{11} + 3 a_{10} x^{10} + 9 c_{15} x^{3} + 9 b_{14} x^{2} + 9 b_{13} x + 3 d_{0}$

Invariants

Residue field characteristic:	$3$
Degree:	$12$
Base field:	$\Q_{3}$
Ramification index $e$:	$12$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$21$
Artin slopes:	$[\frac{9}{4}]$
Swan slopes:	$[\frac{5}{4}]$
Means:	$\langle\frac{5}{6}\rangle$
Rams:	$(5)$
Field count:	$114$ (complete)
Ambiguity:	$6$
Mass:	$54$
Absolute Mass:	$54$

Diagrams

Varying

Indices of inseparability:	$[10,0]$
Associated inertia:	$[2,1]$ (show 84), $[2,2]$ (show 30)
Jump Set:	undefined (show 57), $[2,14]$ (show 57)

Galois groups and Hidden Artin slopes

Select desired size of Galois group.

		Galois groups of order 72
		$\SOPlus(4,2)$ (as 12T35)	$\SOPlus(4,2)$ (as 12T36)	Total
hidden slopes	$[\frac{9}{4}]^{2}$	6	6	12

		Galois groups of order 216
		$S_3^2:S_3$ (as 12T116)	$C_3^2:D_{12}$ (as 12T118)	$S_3^2:S_3$ (as 12T120)	$S_3^2:C_6$ (as 12T121)	Total
hidden slopes	$[2,\frac{9}{4}]^{2}$		6		18	24
		$[\frac{3}{2},\frac{9}{4}]^{2}$	6		6		12
	$[\frac{9}{4}]^{6}$					6	6
		Total	6	6	6	24	42

		Galois groups of order 648
		$C_3\wr D_4$ (as 12T167)	$C_3^3:D_{12}$ (as 12T169)	Total
hidden slopes	$[\frac{3}{2},2,\frac{9}{4}]^{2}$	36	12	48
		$[\frac{3}{2},\frac{9}{4}]^{6}$	12		12
	Total		48	12	60

Fields

Showing 1-50 of 114

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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
3.1.12.21a1.1	6	$x^{12} + 6 x^{10} + 3$	$\SOPlus(4,2)$ (as 12T36)	$72$	$2$	$[\frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[\frac{9}{4}]^{2}$	$[\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 2]$	$z^9 + z^6 + 1,z^2 + 1$	$[2, 14]$
3.1.12.21a1.2	6	$x^{12} + 6 x^{10} + 9 x^{2} + 3$	$\SOPlus(4,2)$ (as 12T36)	$72$	$2$	$[\frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[\frac{9}{4}]^{2}$	$[\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 2]$	$z^9 + z^6 + 1,z^2 + 1$	$[2, 14]$
3.1.12.21a1.3	6	$x^{12} + 6 x^{10} + 18 x^{2} + 3$	$\SOPlus(4,2)$ (as 12T36)	$72$	$2$	$[\frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[\frac{9}{4}]^{2}$	$[\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 2]$	$z^9 + z^6 + 1,z^2 + 1$	$[2, 14]$
3.1.12.21a1.4	6	$x^{12} + 6 x^{10} + 9 x + 3$	$S_3^2:S_3$ (as 12T120)	$216$	$1$	$[\frac{3}{2}, \frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[\frac{1}{2},\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[\frac{3}{2},\frac{9}{4}]^{2}$	$[\frac{1}{2},\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 2]$	$z^9 + z^6 + 1,z^2 + 1$	$[2, 14]$
3.1.12.21a1.5	6	$x^{12} + 6 x^{10} + 9 x^{2} + 9 x + 3$	$S_3^2:S_3$ (as 12T120)	$216$	$1$	$[\frac{3}{2}, \frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[\frac{1}{2},\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[\frac{3}{2},\frac{9}{4}]^{2}$	$[\frac{1}{2},\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 2]$	$z^9 + z^6 + 1,z^2 + 1$	$[2, 14]$
3.1.12.21a1.6	6	$x^{12} + 6 x^{10} + 18 x^{2} + 9 x + 3$	$S_3^2:S_3$ (as 12T120)	$216$	$1$	$[\frac{3}{2}, \frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[\frac{1}{2},\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[\frac{3}{2},\frac{9}{4}]^{2}$	$[\frac{1}{2},\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 2]$	$z^9 + z^6 + 1,z^2 + 1$	$[2, 14]$
3.1.12.21a1.7	12	$x^{12} + 3 x^{11} + 6 x^{10} + 3$	$C_3^3:D_{12}$ (as 12T169)	$648$	$1$	$[\frac{3}{2}, 2, \frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[\frac{1}{2},1,\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[\frac{3}{2},2,\frac{9}{4}]^{2}$	$[\frac{1}{2},1,\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 2]$	$z^9 + z^6 + 1,z^2 + 1$	$[2, 14]$
3.1.12.21a1.8	12	$x^{12} + 3 x^{11} + 6 x^{10} + 9 x^{2} + 3$	$C_3^3:D_{12}$ (as 12T169)	$648$	$1$	$[\frac{3}{2}, 2, \frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[\frac{1}{2},1,\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[\frac{3}{2},2,\frac{9}{4}]^{2}$	$[\frac{1}{2},1,\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 2]$	$z^9 + z^6 + 1,z^2 + 1$	$[2, 14]$
3.1.12.21a1.9	12	$x^{12} + 3 x^{11} + 6 x^{10} + 18 x^{2} + 3$	$C_3^3:D_{12}$ (as 12T169)	$648$	$1$	$[\frac{3}{2}, 2, \frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[\frac{1}{2},1,\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[\frac{3}{2},2,\frac{9}{4}]^{2}$	$[\frac{1}{2},1,\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 2]$	$z^9 + z^6 + 1,z^2 + 1$	$[2, 14]$
3.1.12.21a1.10	12	$x^{12} + 3 x^{11} + 6 x^{10} + 9 x + 3$	$C_3^3:D_{12}$ (as 12T169)	$648$	$1$	$[\frac{3}{2}, 2, \frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[\frac{1}{2},1,\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[\frac{3}{2},2,\frac{9}{4}]^{2}$	$[\frac{1}{2},1,\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 2]$	$z^9 + z^6 + 1,z^2 + 1$	$[2, 14]$
3.1.12.21a1.11	12	$x^{12} + 3 x^{11} + 6 x^{10} + 9 x^{2} + 9 x + 3$	$C_3^3:D_{12}$ (as 12T169)	$648$	$1$	$[\frac{3}{2}, 2, \frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[\frac{1}{2},1,\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[\frac{3}{2},2,\frac{9}{4}]^{2}$	$[\frac{1}{2},1,\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 2]$	$z^9 + z^6 + 1,z^2 + 1$	$[2, 14]$
3.1.12.21a1.12	12	$x^{12} + 3 x^{11} + 6 x^{10} + 18 x^{2} + 9 x + 3$	$C_3^3:D_{12}$ (as 12T169)	$648$	$1$	$[\frac{3}{2}, 2, \frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[\frac{1}{2},1,\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[\frac{3}{2},2,\frac{9}{4}]^{2}$	$[\frac{1}{2},1,\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 2]$	$z^9 + z^6 + 1,z^2 + 1$	$[2, 14]$
3.1.12.21a1.13	6	$x^{12} + 6 x^{11} + 6 x^{10} + 9 x + 3$	$C_3^2:D_{12}$ (as 12T118)	$216$	$1$	$[2, \frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[1,\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[2,\frac{9}{4}]^{2}$	$[1,\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 2]$	$z^9 + z^6 + 1,z^2 + 1$	$[2, 14]$
3.1.12.21a1.14	6	$x^{12} + 6 x^{11} + 6 x^{10} + 9 x^{2} + 9 x + 3$	$C_3^2:D_{12}$ (as 12T118)	$216$	$1$	$[2, \frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[1,\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[2,\frac{9}{4}]^{2}$	$[1,\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 2]$	$z^9 + z^6 + 1,z^2 + 1$	$[2, 14]$
3.1.12.21a1.15	6	$x^{12} + 6 x^{11} + 6 x^{10} + 18 x^{2} + 9 x + 3$	$C_3^2:D_{12}$ (as 12T118)	$216$	$1$	$[2, \frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[1,\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[2,\frac{9}{4}]^{2}$	$[1,\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 2]$	$z^9 + z^6 + 1,z^2 + 1$	$[2, 14]$
3.1.12.21a1.16	6	$x^{12} + 3 x^{10} + 6$	$\SOPlus(4,2)$ (as 12T36)	$72$	$2$	$[\frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[\frac{9}{4}]^{2}$	$[\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 2]$	$z^9 + z^6 + 1,z^2 + 1$	undefined
3.1.12.21a1.17	6	$x^{12} + 3 x^{10} + 9 x^{2} + 6$	$\SOPlus(4,2)$ (as 12T36)	$72$	$2$	$[\frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[\frac{9}{4}]^{2}$	$[\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 2]$	$z^9 + z^6 + 1,z^2 + 1$	undefined
3.1.12.21a1.18	6	$x^{12} + 3 x^{10} + 18 x^{2} + 6$	$\SOPlus(4,2)$ (as 12T36)	$72$	$2$	$[\frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[\frac{9}{4}]^{2}$	$[\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 2]$	$z^9 + z^6 + 1,z^2 + 1$	undefined
3.1.12.21a1.19	6	$x^{12} + 3 x^{10} + 9 x + 6$	$S_3^2:S_3$ (as 12T120)	$216$	$1$	$[\frac{3}{2}, \frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[\frac{1}{2},\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[\frac{3}{2},\frac{9}{4}]^{2}$	$[\frac{1}{2},\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 2]$	$z^9 + z^6 + 1,z^2 + 1$	undefined
3.1.12.21a1.20	6	$x^{12} + 3 x^{10} + 9 x^{2} + 9 x + 6$	$S_3^2:S_3$ (as 12T120)	$216$	$1$	$[\frac{3}{2}, \frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[\frac{1}{2},\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[\frac{3}{2},\frac{9}{4}]^{2}$	$[\frac{1}{2},\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 2]$	$z^9 + z^6 + 1,z^2 + 1$	undefined
3.1.12.21a1.21	6	$x^{12} + 3 x^{10} + 18 x^{2} + 9 x + 6$	$S_3^2:S_3$ (as 12T120)	$216$	$1$	$[\frac{3}{2}, \frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[\frac{1}{2},\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[\frac{3}{2},\frac{9}{4}]^{2}$	$[\frac{1}{2},\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 2]$	$z^9 + z^6 + 1,z^2 + 1$	undefined
3.1.12.21a1.22	12	$x^{12} + 3 x^{11} + 3 x^{10} + 6$	$C_3^3:D_{12}$ (as 12T169)	$648$	$1$	$[\frac{3}{2}, 2, \frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[\frac{1}{2},1,\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[\frac{3}{2},2,\frac{9}{4}]^{2}$	$[\frac{1}{2},1,\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 2]$	$z^9 + z^6 + 1,z^2 + 1$	undefined
3.1.12.21a1.23	12	$x^{12} + 3 x^{11} + 3 x^{10} + 9 x^{2} + 6$	$C_3^3:D_{12}$ (as 12T169)	$648$	$1$	$[\frac{3}{2}, 2, \frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[\frac{1}{2},1,\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[\frac{3}{2},2,\frac{9}{4}]^{2}$	$[\frac{1}{2},1,\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 2]$	$z^9 + z^6 + 1,z^2 + 1$	undefined
3.1.12.21a1.24	12	$x^{12} + 3 x^{11} + 3 x^{10} + 18 x^{2} + 6$	$C_3^3:D_{12}$ (as 12T169)	$648$	$1$	$[\frac{3}{2}, 2, \frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[\frac{1}{2},1,\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[\frac{3}{2},2,\frac{9}{4}]^{2}$	$[\frac{1}{2},1,\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 2]$	$z^9 + z^6 + 1,z^2 + 1$	undefined
3.1.12.21a1.25	6	$x^{12} + 3 x^{11} + 3 x^{10} + 9 x + 6$	$C_3^2:D_{12}$ (as 12T118)	$216$	$1$	$[2, \frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[1,\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[2,\frac{9}{4}]^{2}$	$[1,\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 2]$	$z^9 + z^6 + 1,z^2 + 1$	undefined
3.1.12.21a1.26	6	$x^{12} + 3 x^{11} + 3 x^{10} + 9 x^{2} + 9 x + 6$	$C_3^2:D_{12}$ (as 12T118)	$216$	$1$	$[2, \frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[1,\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[2,\frac{9}{4}]^{2}$	$[1,\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 2]$	$z^9 + z^6 + 1,z^2 + 1$	undefined
3.1.12.21a1.27	6	$x^{12} + 3 x^{11} + 3 x^{10} + 18 x^{2} + 9 x + 6$	$C_3^2:D_{12}$ (as 12T118)	$216$	$1$	$[2, \frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[1,\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[2,\frac{9}{4}]^{2}$	$[1,\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 2]$	$z^9 + z^6 + 1,z^2 + 1$	undefined
3.1.12.21a1.28	12	$x^{12} + 6 x^{11} + 3 x^{10} + 9 x + 6$	$C_3^3:D_{12}$ (as 12T169)	$648$	$1$	$[\frac{3}{2}, 2, \frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[\frac{1}{2},1,\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[\frac{3}{2},2,\frac{9}{4}]^{2}$	$[\frac{1}{2},1,\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 2]$	$z^9 + z^6 + 1,z^2 + 1$	undefined
3.1.12.21a1.29	12	$x^{12} + 6 x^{11} + 3 x^{10} + 9 x^{2} + 9 x + 6$	$C_3^3:D_{12}$ (as 12T169)	$648$	$1$	$[\frac{3}{2}, 2, \frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[\frac{1}{2},1,\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[\frac{3}{2},2,\frac{9}{4}]^{2}$	$[\frac{1}{2},1,\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 2]$	$z^9 + z^6 + 1,z^2 + 1$	undefined
3.1.12.21a1.30	12	$x^{12} + 6 x^{11} + 3 x^{10} + 18 x^{2} + 9 x + 6$	$C_3^3:D_{12}$ (as 12T169)	$648$	$1$	$[\frac{3}{2}, 2, \frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[\frac{1}{2},1,\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[\frac{3}{2},2,\frac{9}{4}]^{2}$	$[\frac{1}{2},1,\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 2]$	$z^9 + z^6 + 1,z^2 + 1$	undefined
3.1.12.21a2.1	6	$x^{12} + 3 x^{10} + 3$	$\SOPlus(4,2)$ (as 12T35)	$72$	$6$	$[\frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[\frac{9}{4}]^{2}$	$[\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 1]$	$z^9 + z^6 + 1,z^2 + 2$	$[2, 14]$
3.1.12.21a2.2	6	$x^{12} + 3 x^{10} + 9 x^{3} + 3$	$S_3^2:C_6$ (as 12T121)	$216$	$3$	$[\frac{9}{4}, \frac{9}{4}]_{4}^{6}$	$[\frac{5}{4},\frac{5}{4}]_{4}^{6}$	$[\frac{9}{4}]^{6}$	$[\frac{5}{4}]^{6}$	$[10, 0]$	$[2, 1]$	$z^9 + z^6 + 1,z^2 + 2$	$[2, 14]$
3.1.12.21a2.3	6	$x^{12} + 3 x^{10} + 9 x^{2} + 3$	$\SOPlus(4,2)$ (as 12T35)	$72$	$6$	$[\frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[\frac{9}{4}]^{2}$	$[\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 1]$	$z^9 + z^6 + 1,z^2 + 2$	$[2, 14]$
3.1.12.21a2.4	6	$x^{12} + 3 x^{10} + 9 x^{3} + 9 x^{2} + 3$	$S_3^2:C_6$ (as 12T121)	$216$	$3$	$[\frac{9}{4}, \frac{9}{4}]_{4}^{6}$	$[\frac{5}{4},\frac{5}{4}]_{4}^{6}$	$[\frac{9}{4}]^{6}$	$[\frac{5}{4}]^{6}$	$[10, 0]$	$[2, 1]$	$z^9 + z^6 + 1,z^2 + 2$	$[2, 14]$
3.1.12.21a2.5	6	$x^{12} + 3 x^{10} + 18 x^{2} + 3$	$\SOPlus(4,2)$ (as 12T35)	$72$	$6$	$[\frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[\frac{9}{4}]^{2}$	$[\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 1]$	$z^9 + z^6 + 1,z^2 + 2$	$[2, 14]$
3.1.12.21a2.6	6	$x^{12} + 3 x^{10} + 9 x^{3} + 18 x^{2} + 3$	$S_3^2:C_6$ (as 12T121)	$216$	$3$	$[\frac{9}{4}, \frac{9}{4}]_{4}^{6}$	$[\frac{5}{4},\frac{5}{4}]_{4}^{6}$	$[\frac{9}{4}]^{6}$	$[\frac{5}{4}]^{6}$	$[10, 0]$	$[2, 1]$	$z^9 + z^6 + 1,z^2 + 2$	$[2, 14]$
3.1.12.21a2.7	6	$x^{12} + 3 x^{10} + 9 x + 3$	$S_3^2:S_3$ (as 12T116)	$216$	$3$	$[\frac{3}{2}, \frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[\frac{1}{2},\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[\frac{3}{2},\frac{9}{4}]^{2}$	$[\frac{1}{2},\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 1]$	$z^9 + z^6 + 1,z^2 + 2$	$[2, 14]$
3.1.12.21a2.8	12	$x^{12} + 3 x^{10} + 9 x^{3} + 9 x + 3$	$C_3\wr D_4$ (as 12T167)	$648$	$3$	$[\frac{3}{2}, \frac{9}{4}, \frac{9}{4}]_{4}^{6}$	$[\frac{1}{2},\frac{5}{4},\frac{5}{4}]_{4}^{6}$	$[\frac{3}{2},\frac{9}{4}]^{6}$	$[\frac{1}{2},\frac{5}{4}]^{6}$	$[10, 0]$	$[2, 1]$	$z^9 + z^6 + 1,z^2 + 2$	$[2, 14]$
3.1.12.21a2.9	12	$x^{12} + 3 x^{10} + 18 x^{3} + 9 x + 3$	$C_3\wr D_4$ (as 12T167)	$648$	$3$	$[\frac{3}{2}, \frac{9}{4}, \frac{9}{4}]_{4}^{6}$	$[\frac{1}{2},\frac{5}{4},\frac{5}{4}]_{4}^{6}$	$[\frac{3}{2},\frac{9}{4}]^{6}$	$[\frac{1}{2},\frac{5}{4}]^{6}$	$[10, 0]$	$[2, 1]$	$z^9 + z^6 + 1,z^2 + 2$	$[2, 14]$
3.1.12.21a2.10	6	$x^{12} + 3 x^{10} + 9 x^{2} + 9 x + 3$	$S_3^2:S_3$ (as 12T116)	$216$	$3$	$[\frac{3}{2}, \frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[\frac{1}{2},\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[\frac{3}{2},\frac{9}{4}]^{2}$	$[\frac{1}{2},\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 1]$	$z^9 + z^6 + 1,z^2 + 2$	$[2, 14]$
3.1.12.21a2.11	12	$x^{12} + 3 x^{10} + 9 x^{3} + 9 x^{2} + 9 x + 3$	$C_3\wr D_4$ (as 12T167)	$648$	$3$	$[\frac{3}{2}, \frac{9}{4}, \frac{9}{4}]_{4}^{6}$	$[\frac{1}{2},\frac{5}{4},\frac{5}{4}]_{4}^{6}$	$[\frac{3}{2},\frac{9}{4}]^{6}$	$[\frac{1}{2},\frac{5}{4}]^{6}$	$[10, 0]$	$[2, 1]$	$z^9 + z^6 + 1,z^2 + 2$	$[2, 14]$
3.1.12.21a2.12	12	$x^{12} + 3 x^{10} + 18 x^{3} + 9 x^{2} + 9 x + 3$	$C_3\wr D_4$ (as 12T167)	$648$	$3$	$[\frac{3}{2}, \frac{9}{4}, \frac{9}{4}]_{4}^{6}$	$[\frac{1}{2},\frac{5}{4},\frac{5}{4}]_{4}^{6}$	$[\frac{3}{2},\frac{9}{4}]^{6}$	$[\frac{1}{2},\frac{5}{4}]^{6}$	$[10, 0]$	$[2, 1]$	$z^9 + z^6 + 1,z^2 + 2$	$[2, 14]$
3.1.12.21a2.13	6	$x^{12} + 3 x^{10} + 18 x^{2} + 9 x + 3$	$S_3^2:S_3$ (as 12T116)	$216$	$3$	$[\frac{3}{2}, \frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[\frac{1}{2},\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[\frac{3}{2},\frac{9}{4}]^{2}$	$[\frac{1}{2},\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 1]$	$z^9 + z^6 + 1,z^2 + 2$	$[2, 14]$
3.1.12.21a2.14	12	$x^{12} + 3 x^{10} + 9 x^{3} + 18 x^{2} + 9 x + 3$	$C_3\wr D_4$ (as 12T167)	$648$	$3$	$[\frac{3}{2}, \frac{9}{4}, \frac{9}{4}]_{4}^{6}$	$[\frac{1}{2},\frac{5}{4},\frac{5}{4}]_{4}^{6}$	$[\frac{3}{2},\frac{9}{4}]^{6}$	$[\frac{1}{2},\frac{5}{4}]^{6}$	$[10, 0]$	$[2, 1]$	$z^9 + z^6 + 1,z^2 + 2$	$[2, 14]$
3.1.12.21a2.15	12	$x^{12} + 3 x^{10} + 18 x^{3} + 18 x^{2} + 9 x + 3$	$C_3\wr D_4$ (as 12T167)	$648$	$3$	$[\frac{3}{2}, \frac{9}{4}, \frac{9}{4}]_{4}^{6}$	$[\frac{1}{2},\frac{5}{4},\frac{5}{4}]_{4}^{6}$	$[\frac{3}{2},\frac{9}{4}]^{6}$	$[\frac{1}{2},\frac{5}{4}]^{6}$	$[10, 0]$	$[2, 1]$	$z^9 + z^6 + 1,z^2 + 2$	$[2, 14]$
3.1.12.21a2.16	36	$x^{12} + 3 x^{11} + 3 x^{10} + 3$	$C_3\wr D_4$ (as 12T167)	$648$	$3$	$[\frac{3}{2}, 2, \frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[\frac{1}{2},1,\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[\frac{3}{2},2,\frac{9}{4}]^{2}$	$[\frac{1}{2},1,\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 1]$	$z^9 + z^6 + 1,z^2 + 2$	$[2, 14]$
3.1.12.21a2.17	36	$x^{12} + 3 x^{11} + 3 x^{10} + 9 x^{3} + 3$	$C_3\wr D_4$ (as 12T167)	$648$	$3$	$[\frac{3}{2}, 2, \frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[\frac{1}{2},1,\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[\frac{3}{2},2,\frac{9}{4}]^{2}$	$[\frac{1}{2},1,\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 1]$	$z^9 + z^6 + 1,z^2 + 2$	$[2, 14]$
3.1.12.21a2.18	36	$x^{12} + 3 x^{11} + 3 x^{10} + 9 x^{2} + 3$	$C_3\wr D_4$ (as 12T167)	$648$	$3$	$[\frac{3}{2}, 2, \frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[\frac{1}{2},1,\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[\frac{3}{2},2,\frac{9}{4}]^{2}$	$[\frac{1}{2},1,\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 1]$	$z^9 + z^6 + 1,z^2 + 2$	$[2, 14]$
3.1.12.21a2.19	36	$x^{12} + 3 x^{11} + 3 x^{10} + 9 x^{3} + 9 x^{2} + 3$	$C_3\wr D_4$ (as 12T167)	$648$	$3$	$[\frac{3}{2}, 2, \frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[\frac{1}{2},1,\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[\frac{3}{2},2,\frac{9}{4}]^{2}$	$[\frac{1}{2},1,\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 1]$	$z^9 + z^6 + 1,z^2 + 2$	$[2, 14]$
3.1.12.21a2.20	36	$x^{12} + 3 x^{11} + 3 x^{10} + 18 x^{2} + 3$	$C_3\wr D_4$ (as 12T167)	$648$	$3$	$[\frac{3}{2}, 2, \frac{9}{4}, \frac{9}{4}]_{4}^{2}$	$[\frac{1}{2},1,\frac{5}{4},\frac{5}{4}]_{4}^{2}$	$[\frac{3}{2},2,\frac{9}{4}]^{2}$	$[\frac{1}{2},1,\frac{5}{4}]^{2}$	$[10, 0]$	$[2, 1]$	$z^9 + z^6 + 1,z^2 + 2$	$[2, 14]$

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