$p$-adic family 2.3.6.24a

• Label: 2.3.6.24a
• Base: 2.1.1.0a1.1
• Degree: \(18\)
• e: \(6\)
• f: \(3\)
• c: \(24\)

Defining polynomial over unramified subextension

$x^{6} + 2 b_{5} x^{5} + 2 a_{3} x^{3} + 4 c_{6} + 2$

Invariants

Residue field characteristic:	$2$
Degree:	$18$
Base field:	$\Q_{2}$
Ramification index $e$:	$6$
Residue field degree $f$:	$3$
Discriminant exponent $c$:	$24$
Artin slopes:	$[2]$
Swan slopes:	$[1]$
Means:	$\langle\frac{1}{2}\rangle$
Rams:	$(3)$
Field count:	$40$ (complete)
Ambiguity:	$6$
Mass:	$56$
Absolute Mass:	$56/3$

Diagrams

Varying

Indices of inseparability:	$[3,0]$
Associated inertia:	$[2,1]$
Jump Set:	$[3,6]$ (show 1), $[3,9]$ (show 32), $[3,11]$ (show 6), $[3,12]$ (show 1)

Galois groups and Hidden Artin slopes

Select desired size of Galois group.

		Galois groups of order 36
		$C_6\times S_3$ (as 18T6)
hidden slopes	$[\ ]^{2}$	2

		Galois groups of order 72
		$S_3\times A_4$ (as 18T31)	$S_3\times A_4$ (as 18T32)	Total
hidden slopes	$[2]^{2}$	1	1	2

		Galois groups of order 144
		$A_4\times D_6$ (as 18T60)	$C_6\times S_4$ (as 18T61)	Total
hidden slopes	$[2,2]^{2}$	2		2
		$[\frac{4}{3},\frac{4}{3}]^{2}$		2	2
	Total		2	2	4

		Galois groups of order 288
		$A_4\times S_4$ (as 18T114)	$A_4\times S_4$ (as 18T115)	Total
hidden slopes	$[\frac{4}{3},\frac{4}{3},2]^{2}$	1	1	2

		Galois groups of order 576
		$A_4^2:C_2^2$ (as 18T175)	$A_4^2:C_2^2$ (as 18T176)	Total
hidden slopes	$[\frac{4}{3},\frac{4}{3},2,2]^{2}$		2	2
		$[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3}]^{2}$	2		2
	Total		2	2	4

		Galois groups of order 1152
		$C_2^4:(S_3\times A_4)$ (as 18T268)	$C_2^4:(S_3\times A_4)$ (as 18T271)	Total
hidden slopes	$[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2]^{2}$	3	3	6

		Galois groups of order 2304
		$C_2^4:(A_4\times D_6)$ (as 18T366)	$C_2^4:(C_6\times S_4)$ (as 18T367)	Total
hidden slopes	$[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,2]^{2}$	6		6
		$[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3}]^{2}$		2	2
	Total		6	2	8

		Galois groups of order 4608
		$C_2^4:(A_4\times S_4)$ (as 18T462)	$C_2^4:(A_4\times S_4)$ (as 18T463)	Total
hidden slopes	$[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2]^{2}$	3	3	6

		Galois groups of order 9216
		$C_2^5.(A_4\times S_4)$ (as 18T544)
hidden slopes	$[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,2]^{2}$	6

Fields

Showing all 40

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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
2.3.6.24a1.1	2	$( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{3} + 2$	$S_3 \times C_6$ (as 18T6)	$36$	$6$	$[2]_{3}^{6}$	$[1]_{3}^{6}$	$[\ ]^{2}$	$[\ ]^{2}$	$[3, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + (t^2 + t)$	$[3, 6]$
2.3.6.24a1.2	2	$( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{3} + 6$	$S_3 \times C_6$ (as 18T6)	$36$	$6$	$[2]_{3}^{6}$	$[1]_{3}^{6}$	$[\ ]^{2}$	$[\ ]^{2}$	$[3, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + (t^2 + t)$	$[3, 12]$
2.3.6.24a1.3	2	$( x^{3} + x + 1 )^{6} + 2 x ( x^{3} + x + 1 )^{5} + 2 ( x^{3} + x + 1 )^{3} + 2$	$A_4^2:C_2^2$ (as 18T175)	$576$	$2$	$[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2]_{3}^{6}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1]_{3}^{6}$	$[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3}]^{2}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3}]^{2}$	$[3, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + (t^2 + t)$	$[3, 11]$
2.3.6.24a1.4	2	$( x^{3} + x + 1 )^{6} + 2 x ( x^{3} + x + 1 )^{5} + 2 ( x^{3} + x + 1 )^{3} + 6$	$A_4^2:C_2^2$ (as 18T175)	$576$	$2$	$[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2]_{3}^{6}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1]_{3}^{6}$	$[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3}]^{2}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3}]^{2}$	$[3, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + (t^2 + t)$	$[3, 11]$
2.3.6.24a1.5	2	$( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 2 ( x^{3} + x + 1 )^{3} + 2$	$C_6\times S_4$ (as 18T61)	$144$	$6$	$[\frac{4}{3}, \frac{4}{3}, 2]_{3}^{6}$	$[\frac{1}{3},\frac{1}{3},1]_{3}^{6}$	$[\frac{4}{3},\frac{4}{3}]^{2}$	$[\frac{1}{3},\frac{1}{3}]^{2}$	$[3, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + (t^2 + t)$	$[3, 11]$
2.3.6.24a1.6	2	$( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 2 ( x^{3} + x + 1 )^{3} + 6$	$C_6\times S_4$ (as 18T61)	$144$	$6$	$[\frac{4}{3}, \frac{4}{3}, 2]_{3}^{6}$	$[\frac{1}{3},\frac{1}{3},1]_{3}^{6}$	$[\frac{4}{3},\frac{4}{3}]^{2}$	$[\frac{1}{3},\frac{1}{3}]^{2}$	$[3, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + (t^2 + t)$	$[3, 11]$
2.3.6.24a1.7	2	$( x^{3} + x + 1 )^{6} + \left(2 x + 2\right) ( x^{3} + x + 1 )^{5} + 2 ( x^{3} + x + 1 )^{3} + 2$	$C_2^4:(C_6\times S_4)$ (as 18T367)	$2304$	$2$	$[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2]_{3}^{6}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1]_{3}^{6}$	$[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3}]^{2}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3}]^{2}$	$[3, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + (t^2 + t)$	$[3, 11]$
2.3.6.24a1.8	2	$( x^{3} + x + 1 )^{6} + \left(2 x + 2\right) ( x^{3} + x + 1 )^{5} + 2 ( x^{3} + x + 1 )^{3} + 6$	$C_2^4:(C_6\times S_4)$ (as 18T367)	$2304$	$2$	$[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2]_{3}^{6}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1]_{3}^{6}$	$[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3}]^{2}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3}]^{2}$	$[3, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + (t^2 + t)$	$[3, 11]$
2.3.6.24a2.1	6	$( x^{3} + x + 1 )^{6} + 2 x ( x^{3} + x + 1 )^{3} + 2$	$C_2^5.(A_4\times S_4)$ (as 18T544)	$9216$	$2$	$[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, 2]_{3}^{6}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,1]_{3}^{6}$	$[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,2]^{2}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1]^{2}$	$[3, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + (t^2 + t + 1)$	$[3, 9]$
2.3.6.24a2.2	6	$( x^{3} + x + 1 )^{6} + 2 x ( x^{3} + x + 1 )^{3} + 6$	$C_2^5.(A_4\times S_4)$ (as 18T544)	$9216$	$2$	$[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, 2]_{3}^{6}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,1]_{3}^{6}$	$[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,2]^{2}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1]^{2}$	$[3, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + (t^2 + t + 1)$	$[3, 9]$
2.3.6.24a2.3	6	$( x^{3} + x + 1 )^{6} + 2 x^{2} ( x^{3} + x + 1 )^{5} + 2 x ( x^{3} + x + 1 )^{3} + 2$	$C_2^4:(A_4\times D_6)$ (as 18T366)	$2304$	$2$	$[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, 2]_{3}^{6}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,1]_{3}^{6}$	$[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,2]^{2}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1]^{2}$	$[3, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + (t^2 + t + 1)$	$[3, 9]$
2.3.6.24a2.4	6	$( x^{3} + x + 1 )^{6} + 2 x^{2} ( x^{3} + x + 1 )^{5} + 2 x ( x^{3} + x + 1 )^{3} + 6$	$C_2^4:(A_4\times D_6)$ (as 18T366)	$2304$	$2$	$[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, 2]_{3}^{6}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,1]_{3}^{6}$	$[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,2]^{2}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1]^{2}$	$[3, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + (t^2 + t + 1)$	$[3, 9]$
2.3.6.24a2.5	6	$( x^{3} + x + 1 )^{6} + 2 x ( x^{3} + x + 1 )^{5} + 2 x ( x^{3} + x + 1 )^{3} + 2$	$C_2^4:(A_4\times D_6)$ (as 18T366)	$2304$	$2$	$[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, 2]_{3}^{6}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,1]_{3}^{6}$	$[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,2]^{2}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1]^{2}$	$[3, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + (t^2 + t + 1)$	$[3, 9]$
2.3.6.24a2.6	6	$( x^{3} + x + 1 )^{6} + 2 x ( x^{3} + x + 1 )^{5} + 2 x ( x^{3} + x + 1 )^{3} + 6$	$C_2^4:(A_4\times D_6)$ (as 18T366)	$2304$	$2$	$[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, 2]_{3}^{6}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,1]_{3}^{6}$	$[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,2]^{2}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1]^{2}$	$[3, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + (t^2 + t + 1)$	$[3, 9]$
2.3.6.24a2.7	6	$( x^{3} + x + 1 )^{6} + \left(2 x^{2} + 2 x\right) ( x^{3} + x + 1 )^{5} + 2 x ( x^{3} + x + 1 )^{3} + 2$	$C_2^5.(A_4\times S_4)$ (as 18T544)	$9216$	$2$	$[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, 2]_{3}^{6}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,1]_{3}^{6}$	$[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,2]^{2}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1]^{2}$	$[3, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + (t^2 + t + 1)$	$[3, 9]$
2.3.6.24a2.8	6	$( x^{3} + x + 1 )^{6} + \left(2 x^{2} + 2 x\right) ( x^{3} + x + 1 )^{5} + 2 x ( x^{3} + x + 1 )^{3} + 6$	$C_2^5.(A_4\times S_4)$ (as 18T544)	$9216$	$2$	$[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, 2]_{3}^{6}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,1]_{3}^{6}$	$[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,2]^{2}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1]^{2}$	$[3, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + (t^2 + t + 1)$	$[3, 9]$
2.3.6.24a2.9	2	$( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 2 x ( x^{3} + x + 1 )^{3} + 2$	$A_4\times D_6$ (as 18T60)	$144$	$2$	$[2, 2, 2]_{3}^{6}$	$[1,1,1]_{3}^{6}$	$[2,2]^{2}$	$[1,1]^{2}$	$[3, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + (t^2 + t + 1)$	$[3, 9]$
2.3.6.24a2.10	2	$( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + 2 x ( x^{3} + x + 1 )^{3} + 6$	$A_4\times D_6$ (as 18T60)	$144$	$2$	$[2, 2, 2]_{3}^{6}$	$[1,1,1]_{3}^{6}$	$[2,2]^{2}$	$[1,1]^{2}$	$[3, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + (t^2 + t + 1)$	$[3, 9]$
2.3.6.24a2.11	2	$( x^{3} + x + 1 )^{6} + \left(2 x^{2} + 2\right) ( x^{3} + x + 1 )^{5} + 2 x ( x^{3} + x + 1 )^{3} + 2$	$A_4^2:C_2^2$ (as 18T176)	$576$	$2$	$[\frac{4}{3}, \frac{4}{3}, 2, 2, 2]_{3}^{6}$	$[\frac{1}{3},\frac{1}{3},1,1,1]_{3}^{6}$	$[\frac{4}{3},\frac{4}{3},2,2]^{2}$	$[\frac{1}{3},\frac{1}{3},1,1]^{2}$	$[3, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + (t^2 + t + 1)$	$[3, 9]$
2.3.6.24a2.12	2	$( x^{3} + x + 1 )^{6} + \left(2 x^{2} + 2\right) ( x^{3} + x + 1 )^{5} + 2 x ( x^{3} + x + 1 )^{3} + 6$	$A_4^2:C_2^2$ (as 18T176)	$576$	$2$	$[\frac{4}{3}, \frac{4}{3}, 2, 2, 2]_{3}^{6}$	$[\frac{1}{3},\frac{1}{3},1,1,1]_{3}^{6}$	$[\frac{4}{3},\frac{4}{3},2,2]^{2}$	$[\frac{1}{3},\frac{1}{3},1,1]^{2}$	$[3, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + (t^2 + t + 1)$	$[3, 9]$
2.3.6.24a2.13	6	$( x^{3} + x + 1 )^{6} + \left(2 x + 2\right) ( x^{3} + x + 1 )^{5} + 2 x ( x^{3} + x + 1 )^{3} + 2$	$C_2^5.(A_4\times S_4)$ (as 18T544)	$9216$	$2$	$[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, 2]_{3}^{6}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,1]_{3}^{6}$	$[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,2]^{2}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1]^{2}$	$[3, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + (t^2 + t + 1)$	$[3, 9]$
2.3.6.24a2.14	6	$( x^{3} + x + 1 )^{6} + \left(2 x + 2\right) ( x^{3} + x + 1 )^{5} + 2 x ( x^{3} + x + 1 )^{3} + 6$	$C_2^5.(A_4\times S_4)$ (as 18T544)	$9216$	$2$	$[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, 2]_{3}^{6}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,1]_{3}^{6}$	$[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,2]^{2}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1]^{2}$	$[3, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + (t^2 + t + 1)$	$[3, 9]$
2.3.6.24a2.15	6	$( x^{3} + x + 1 )^{6} + \left(2 x^{2} + 2 x + 2\right) ( x^{3} + x + 1 )^{5} + 2 x ( x^{3} + x + 1 )^{3} + 2$	$C_2^4:(A_4\times D_6)$ (as 18T366)	$2304$	$2$	$[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, 2]_{3}^{6}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,1]_{3}^{6}$	$[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,2]^{2}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1]^{2}$	$[3, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + (t^2 + t + 1)$	$[3, 9]$
2.3.6.24a2.16	6	$( x^{3} + x + 1 )^{6} + \left(2 x^{2} + 2 x + 2\right) ( x^{3} + x + 1 )^{5} + 2 x ( x^{3} + x + 1 )^{3} + 6$	$C_2^4:(A_4\times D_6)$ (as 18T366)	$2304$	$2$	$[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, 2]_{3}^{6}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,1]_{3}^{6}$	$[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2,2]^{2}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1]^{2}$	$[3, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + (t^2 + t + 1)$	$[3, 9]$
2.3.6.24a3.1	3	$( x^{3} + x + 1 )^{6} + \left(2 x + 2\right) ( x^{3} + x + 1 )^{3} + 2$	$C_2^4:(A_4\times S_4)$ (as 18T462)	$4608$	$2$	$[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2]_{3}^{6}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1]_{3}^{6}$	$[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2]^{2}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1]^{2}$	$[3, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + 1$	$[3, 9]$
2.3.6.24a3.2	3	$( x^{3} + x + 1 )^{6} + \left(2 x + 2\right) ( x^{3} + x + 1 )^{3} + 4 x^{2} + 2$	$C_2^4:(A_4\times S_4)$ (as 18T463)	$4608$	$2$	$[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2]_{3}^{6}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1]_{3}^{6}$	$[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2]^{2}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1]^{2}$	$[3, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + 1$	$[3, 9]$
2.3.6.24a3.3	3	$( x^{3} + x + 1 )^{6} + 2 x^{2} ( x^{3} + x + 1 )^{5} + \left(2 x + 2\right) ( x^{3} + x + 1 )^{3} + 2$	$C_2^4:(S_3\times A_4)$ (as 18T271)	$1152$	$2$	$[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2]_{3}^{6}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1]_{3}^{6}$	$[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2]^{2}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1]^{2}$	$[3, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + 1$	$[3, 9]$
2.3.6.24a3.4	3	$( x^{3} + x + 1 )^{6} + 2 x^{2} ( x^{3} + x + 1 )^{5} + \left(2 x + 2\right) ( x^{3} + x + 1 )^{3} + 4 x^{2} + 2$	$C_2^4:(S_3\times A_4)$ (as 18T268)	$1152$	$2$	$[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2]_{3}^{6}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1]_{3}^{6}$	$[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2]^{2}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1]^{2}$	$[3, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + 1$	$[3, 9]$
2.3.6.24a3.5	1	$( x^{3} + x + 1 )^{6} + 2 x ( x^{3} + x + 1 )^{5} + \left(2 x + 2\right) ( x^{3} + x + 1 )^{3} + 2$	$A_4\times S_4$ (as 18T114)	$288$	$2$	$[\frac{4}{3}, \frac{4}{3}, 2, 2]_{3}^{6}$	$[\frac{1}{3},\frac{1}{3},1,1]_{3}^{6}$	$[\frac{4}{3},\frac{4}{3},2]^{2}$	$[\frac{1}{3},\frac{1}{3},1]^{2}$	$[3, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + 1$	$[3, 9]$
2.3.6.24a3.6	1	$( x^{3} + x + 1 )^{6} + 2 x ( x^{3} + x + 1 )^{5} + \left(2 x + 2\right) ( x^{3} + x + 1 )^{3} + 4 x^{2} + 2$	$A_4\times S_4$ (as 18T115)	$288$	$2$	$[\frac{4}{3}, \frac{4}{3}, 2, 2]_{3}^{6}$	$[\frac{1}{3},\frac{1}{3},1,1]_{3}^{6}$	$[\frac{4}{3},\frac{4}{3},2]^{2}$	$[\frac{1}{3},\frac{1}{3},1]^{2}$	$[3, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + 1$	$[3, 9]$
2.3.6.24a3.7	3	$( x^{3} + x + 1 )^{6} + \left(2 x^{2} + 2 x\right) ( x^{3} + x + 1 )^{5} + \left(2 x + 2\right) ( x^{3} + x + 1 )^{3} + 2$	$C_2^4:(S_3\times A_4)$ (as 18T271)	$1152$	$2$	$[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2]_{3}^{6}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1]_{3}^{6}$	$[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2]^{2}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1]^{2}$	$[3, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + 1$	$[3, 9]$
2.3.6.24a3.8	3	$( x^{3} + x + 1 )^{6} + \left(2 x^{2} + 2 x\right) ( x^{3} + x + 1 )^{5} + \left(2 x + 2\right) ( x^{3} + x + 1 )^{3} + 4 x^{2} + 2$	$C_2^4:(S_3\times A_4)$ (as 18T268)	$1152$	$2$	$[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2]_{3}^{6}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1]_{3}^{6}$	$[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2]^{2}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1]^{2}$	$[3, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + 1$	$[3, 9]$
2.3.6.24a3.9	3	$( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + \left(2 x + 2\right) ( x^{3} + x + 1 )^{3} + 2$	$C_2^4:(A_4\times S_4)$ (as 18T462)	$4608$	$2$	$[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2]_{3}^{6}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1]_{3}^{6}$	$[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2]^{2}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1]^{2}$	$[3, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + 1$	$[3, 9]$
2.3.6.24a3.10	3	$( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{5} + \left(2 x + 2\right) ( x^{3} + x + 1 )^{3} + 4 x^{2} + 2$	$C_2^4:(A_4\times S_4)$ (as 18T463)	$4608$	$2$	$[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2]_{3}^{6}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1]_{3}^{6}$	$[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2]^{2}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1]^{2}$	$[3, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + 1$	$[3, 9]$
2.3.6.24a3.11	3	$( x^{3} + x + 1 )^{6} + \left(2 x^{2} + 2\right) ( x^{3} + x + 1 )^{5} + \left(2 x + 2\right) ( x^{3} + x + 1 )^{3} + 2$	$C_2^4:(S_3\times A_4)$ (as 18T271)	$1152$	$2$	$[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2]_{3}^{6}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1]_{3}^{6}$	$[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2]^{2}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1]^{2}$	$[3, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + 1$	$[3, 9]$
2.3.6.24a3.12	3	$( x^{3} + x + 1 )^{6} + \left(2 x^{2} + 2\right) ( x^{3} + x + 1 )^{5} + \left(2 x + 2\right) ( x^{3} + x + 1 )^{3} + 4 x^{2} + 2$	$C_2^4:(S_3\times A_4)$ (as 18T268)	$1152$	$2$	$[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2]_{3}^{6}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1]_{3}^{6}$	$[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2]^{2}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1]^{2}$	$[3, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + 1$	$[3, 9]$
2.3.6.24a3.13	3	$( x^{3} + x + 1 )^{6} + \left(2 x + 2\right) ( x^{3} + x + 1 )^{5} + \left(2 x + 2\right) ( x^{3} + x + 1 )^{3} + 2$	$C_2^4:(A_4\times S_4)$ (as 18T462)	$4608$	$2$	$[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2]_{3}^{6}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1]_{3}^{6}$	$[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2]^{2}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1]^{2}$	$[3, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + 1$	$[3, 9]$
2.3.6.24a3.14	3	$( x^{3} + x + 1 )^{6} + \left(2 x + 2\right) ( x^{3} + x + 1 )^{5} + \left(2 x + 2\right) ( x^{3} + x + 1 )^{3} + 4 x^{2} + 2$	$C_2^4:(A_4\times S_4)$ (as 18T463)	$4608$	$2$	$[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2]_{3}^{6}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1]_{3}^{6}$	$[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},2]^{2}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1]^{2}$	$[3, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + 1$	$[3, 9]$
2.3.6.24a3.15	1	$( x^{3} + x + 1 )^{6} + \left(2 x^{2} + 2 x + 2\right) ( x^{3} + x + 1 )^{5} + \left(2 x + 2\right) ( x^{3} + x + 1 )^{3} + 2$	$S_3\times A_4$ (as 18T32)	$72$	$2$	$[2, 2]_{3}^{6}$	$[1,1]_{3}^{6}$	$[2]^{2}$	$[1]^{2}$	$[3, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + 1$	$[3, 9]$
2.3.6.24a3.16	1	$( x^{3} + x + 1 )^{6} + \left(2 x^{2} + 2 x + 2\right) ( x^{3} + x + 1 )^{5} + \left(2 x + 2\right) ( x^{3} + x + 1 )^{3} + 4 x^{2} + 2$	$S_3\times A_4$ (as 18T31)	$72$	$2$	$[2, 2]_{3}^{6}$	$[1,1]_{3}^{6}$	$[2]^{2}$	$[1]^{2}$	$[3, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + 1$	$[3, 9]$

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