$p$-adic family 2.4.4.44a

• Label: 2.4.4.44a
• Base: 2.1.1.0a1.1
• Degree: \(16\)
• e: \(4\)
• f: \(4\)
• c: \(44\)

Defining polynomial over unramified subextension

$x^{4} + 8 b_{11} x^{3} + 4 b_{6} x^{2} + 8 b_{9} x + 8 c_{8} + 16 c_{12} + 2$

Invariants

Residue field characteristic:	$2$
Degree:	$16$
Base field:	$\Q_{2}$
Ramification index $e$:	$4$
Residue field degree $f$:	$4$
Discriminant exponent $c$:	$44$
Artin slopes:	$[3,4]$
Swan slopes:	$[2,3]$
Means:	$\langle1,2\rangle$
Rams:	$(2,4)$
Field count:	$2158$ (complete)
Ambiguity:	$16$
Mass:	$4096$
Absolute Mass:	$1024$

Diagrams

Varying

Indices of inseparability:	$[8,4,0]$
Associated inertia:	$[1,1]$
Jump Set:	$[1,3,7]$

Galois groups and Hidden Artin slopes

Select desired size of Galois group. Note that the following data has not all been computed for fields in this family, so the tables below are incomplete.

		Galois groups of order 16
		$C_{16}$ (as 16T1)	$C_4^2$ (as 16T4)	$C_2\times C_8$ (as 16T5)	$\OD_{16}$ (as 16T6)	$C_4:C_4$ (as 16T8)	Total
hidden slopes	$[\ ]$	4	2	2	2	2	12

		Galois groups of order 32
		$C_2\times \OD_{16}$ (as 16T15)	$\OD_{16}:C_2$ (as 16T16)	$C_4^2:C_2$ (as 16T17)	$C_4\times D_4$ (as 16T19)	$\OD_{32}$ (as 16T22)	$D_4:C_4$ (as 16T26)	$C_4\wr C_2$ (as 16T28)	Total
hidden slopes	$[2]$	2	4	2	4	4	2	2	20
		$[\ ]^{2}$					2			2
	Total		2	4	2	4	6	2	2	22

		Galois groups of order 64
		$C_2^3.D_4$ (as 16T75)	$\OD_{16}.C_2^2$ (as 16T86)	$\OD_{16}:C_2^2$ (as 16T99)	$\OD_{32}:C_2$ (as 16T104)	$C_2^3.C_2^3$ (as 16T112)	$C_4^2:C_2^2$ (as 16T122)	$C_4^2.C_4$ (as 16T131)	$C_4^2.C_4$ (as 16T139)	$C_4^2:C_4$ (as 16T154)	$C_4^2:C_4$ (as 16T168)	Total
hidden slopes	$[2,2]$	2	2	2	4	4	2					16
		$[2,3]$							2	4	2	4	12
	Total		2	2	2	4	4	2	2	4	2	4	28

		Galois groups of order 128
		$C_2^4.D_4$ (as 16T208)	$(C_2\times C_8):D_4$ (as 16T209)	$(C_2^2\times D_4):C_4$ (as 16T213)	$\OD_{16}:D_4$ (as 16T216)	$C_8.\OD_{16}$ (as 16T220)	$C_2^4.D_4$ (as 16T224)	$C_2^5:C_4$ (as 16T227)	$C_2^4.D_4$ (as 16T236)	$C_4^2.C_8$ (as 16T242)	$(C_2^3\times C_4):C_4$ (as 16T243)	$C_2^4.D_4$ (as 16T247)	$C_2^5:C_4$ (as 16T261)	$(C_2^2\times D_4):C_4$ (as 16T263)	$C_2^4.D_4$ (as 16T268)	$\OD_{16}:D_4$ (as 16T278)	$(C_2\times C_8):D_4$ (as 16T279)	$C_4^2.(C_2\times C_4)$ (as 16T285)	$(C_2^3\times C_4):C_4$ (as 16T292)	$C_2^4.C_8$ (as 16T306)	$(C_2\times C_4^2):C_4$ (as 16T314)	$(C_2^2\times D_4):C_4$ (as 16T315)	$C_2^4.D_4$ (as 16T317)	$C_4^2.(C_2\times C_4)$ (as 16T362)	$(C_4\times C_8).C_4$ (as 16T378)	$(C_4\times C_8):C_4$ (as 16T413)	Total
hidden slopes	$[2,2,2]$													2				2	8	8		4					24
		$[2,2,3]$					8															8			8	4	4	32
	$[2,2,\frac{7}{2}]$		4	4	2	2		2	4	2		4	4	4		2	2	4		4			2	2				48
		$[2,3]^{2}$									8																	8
	Total		4	4	2	2	8	2	4	2	8	4	4	4	2	2	2	4	2	12	8	8	6	2	8	4	4	112

		Galois groups of order 256
		$(C_4\times C_8).D_4$ (as 16T534)	$C_2^4.\OD_{16}$ (as 16T561)	$(C_4\times C_8).C_2^3$ (as 16T577)	$C_2^5.C_8$ (as 16T591)	$D_4^2:C_4$ (as 16T663)	$D_4^2.C_4$ (as 16T671)	$D_4^2:C_4$ (as 16T685)	$D_4^2.C_4$ (as 16T692)	Total
hidden slopes	$[2,2,2,2]$		8		8					16
		$[2,2,2,3]$	8		8						16
	$[2,2,2,\frac{7}{2}]$						8	8	8	8	32
		Total	8	8	8	8	8	8	8	8	64

		Galois groups of order 512
		$C_2^6:(C_2\times C_4)$ (as 16T815)	$C_2^3\wr C_2:C_4$ (as 16T820)	$C_2^6.C_8$ (as 16T841)	$(C_2\times D_4^2):C_4$ (as 16T858)	$(C_2\times D_4^2).C_4$ (as 16T859)	$C_2^6:(C_2\times C_4)$ (as 16T863)	$(C_2\times C_4^3).C_4$ (as 16T888)	$(C_2\times C_4^3):C_4$ (as 16T898)	$C_2^6.C_8$ (as 16T917)	$C_2^3\wr C_2:C_4$ (as 16T939)	Total
hidden slopes	$[2,2,2,2,\frac{7}{2}]$			16	16	16				16		64
		$[2,2,2,3,3]$							32	32			64
	$[2,2,2,\frac{7}{2},\frac{7}{2}]$		16	16				16				16	64
		Total	16	16	16	16	16	16	32	32	16	16	192

		Galois groups of order 1024
		$C_2^4.C_2\wr C_4$ (as 16T1132)	$(C_2^2\times C_4^2).\OD_{16}$ (as 16T1138)	$C_2^6.\OD_{16}$ (as 16T1150)	$C_2^7.C_8$ (as 16T1155)	$C_4^3.(C_2^2\times C_4)$ (as 16T1186)	$C_2^6.\OD_{16}$ (as 16T1204)	$C_2^7.C_8$ (as 16T1223)	$C_2^4.C_2\wr C_4$ (as 16T1239)	$(C_2^2\times C_4^2).\OD_{16}$ (as 16T1243)	$C_2^4.C_2\wr C_4$ (as 16T1269)	Total
hidden slopes	$[2,2,2,2,3,3]$					64						64
		$[2,2,2,2,\frac{7}{2},\frac{7}{2}]$				32			32				64
	$[2,2,2,3,\frac{7}{2},\frac{7}{2}]$		32		32			32		32			128
		$[2,2,2,\frac{5}{2},\frac{7}{2},\frac{7}{2}]$										128	128
	$[2,2,3,3,\frac{7}{2}]_{2}$			32							32		64
		Total	32	32	32	32	64	32	32	32	32	128	448

		Galois groups of order 2048
		$C_2\wr (C_2\times C_4)$ (as 16T1379)
hidden slopes	$[2,2,2,2,\frac{7}{2},\frac{7}{2},\frac{7}{2}]$	32

Fields

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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
2.4.4.44a1.1		$( x^{4} + x + 1 )^{4} + 2$	$C_4 \times D_4$ (as 16T19)	$32$	$8$	$[2, 3, 4]^{4}$	$[1,2,3]^{4}$	$[2]$	$[1]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.2		$( x^{4} + x + 1 )^{4} + 16 x^{3} + 2$	$(C_8:C_2):C_2$ (as 16T16)	$32$	$8$	$[2, 3, 4]^{4}$	$[1,2,3]^{4}$	$[2]$	$[1]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.3		$( x^{4} + x + 1 )^{4} + 8 x^{3} ( x^{4} + x + 1 )^{3} + 2$	$(C_2^2\times D_4):C_4$ (as 16T315)	$128$	$2$	$[2, 2, 2, 3, 4]^{4}$	$[1,1,1,2,3]^{4}$	$[2,2,2]$	$[1,1,1]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.4		$( x^{4} + x + 1 )^{4} + 8 x^{3} ( x^{4} + x + 1 )^{3} + 16 x^{3} + 2$	$(C_2^2\times D_4):C_4$ (as 16T315)	$128$	$2$	$[2, 2, 2, 3, 4]^{4}$	$[1,1,1,2,3]^{4}$	$[2,2,2]$	$[1,1,1]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.5		$( x^{4} + x + 1 )^{4} + 8 x ( x^{4} + x + 1 )^{3} + 2$	$C_2^3.C_2^3$ (as 16T112)	$64$	$4$	$[2, 2, 3, 4]^{4}$	$[1,1,2,3]^{4}$	$[2,2]$	$[1,1]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.6		$( x^{4} + x + 1 )^{4} + 8 x ( x^{4} + x + 1 )^{3} + 16 x^{3} + 2$	$\OD_{16}:C_2^2$ (as 16T99)	$64$	$4$	$[2, 2, 3, 4]^{4}$	$[1,1,2,3]^{4}$	$[2,2]$	$[1,1]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.7		$( x^{4} + x + 1 )^{4} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 )^{3} + 2$	$C_4 \times D_4$ (as 16T19)	$32$	$8$	$[2, 3, 4]^{4}$	$[1,2,3]^{4}$	$[2]$	$[1]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.8		$( x^{4} + x + 1 )^{4} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 )^{3} + 16 x^{3} + 2$	$(C_8:C_2):C_2$ (as 16T16)	$32$	$8$	$[2, 3, 4]^{4}$	$[1,2,3]^{4}$	$[2]$	$[1]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.9		$( x^{4} + x + 1 )^{4} + 8 x^{3} ( x^{4} + x + 1 ) + 2$	$C_2\wr (C_2\times C_4)$ (as 16T1379)	$2048$	$2$	$[2, 2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}, 4]^{4}$	$[1,1,1,1,2,\frac{5}{2},\frac{5}{2},\frac{5}{2},3]^{4}$	$[2,2,2,2,\frac{7}{2},\frac{7}{2},\frac{7}{2}]$	$[1,1,1,1,\frac{5}{2},\frac{5}{2},\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.10		$( x^{4} + x + 1 )^{4} + 8 x^{3} ( x^{4} + x + 1 )^{3} + 8 x^{3} ( x^{4} + x + 1 ) + 2$	$C_2\wr (C_2\times C_4)$ (as 16T1379)	$2048$	$2$	$[2, 2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}, 4]^{4}$	$[1,1,1,1,2,\frac{5}{2},\frac{5}{2},\frac{5}{2},3]^{4}$	$[2,2,2,2,\frac{7}{2},\frac{7}{2},\frac{7}{2}]$	$[1,1,1,1,\frac{5}{2},\frac{5}{2},\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.11		$( x^{4} + x + 1 )^{4} + 8 x^{2} ( x^{4} + x + 1 )^{3} + 8 x^{3} ( x^{4} + x + 1 ) + 2$	$C_2\wr (C_2\times C_4)$ (as 16T1379)	$2048$	$2$	$[2, 2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}, 4]^{4}$	$[1,1,1,1,2,\frac{5}{2},\frac{5}{2},\frac{5}{2},3]^{4}$	$[2,2,2,2,\frac{7}{2},\frac{7}{2},\frac{7}{2}]$	$[1,1,1,1,\frac{5}{2},\frac{5}{2},\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.12		$( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x^{2}\right) ( x^{4} + x + 1 )^{3} + 8 x^{3} ( x^{4} + x + 1 ) + 2$	$C_2\wr (C_2\times C_4)$ (as 16T1379)	$2048$	$2$	$[2, 2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}, 4]^{4}$	$[1,1,1,1,2,\frac{5}{2},\frac{5}{2},\frac{5}{2},3]^{4}$	$[2,2,2,2,\frac{7}{2},\frac{7}{2},\frac{7}{2}]$	$[1,1,1,1,\frac{5}{2},\frac{5}{2},\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.13		$( x^{4} + x + 1 )^{4} + 8 x ( x^{4} + x + 1 )^{3} + 8 x^{3} ( x^{4} + x + 1 ) + 2$	$C_2\wr (C_2\times C_4)$ (as 16T1379)	$2048$	$2$	$[2, 2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}, 4]^{4}$	$[1,1,1,1,2,\frac{5}{2},\frac{5}{2},\frac{5}{2},3]^{4}$	$[2,2,2,2,\frac{7}{2},\frac{7}{2},\frac{7}{2}]$	$[1,1,1,1,\frac{5}{2},\frac{5}{2},\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.14		$( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x\right) ( x^{4} + x + 1 )^{3} + 8 x^{3} ( x^{4} + x + 1 ) + 2$	$C_2\wr (C_2\times C_4)$ (as 16T1379)	$2048$	$2$	$[2, 2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}, 4]^{4}$	$[1,1,1,1,2,\frac{5}{2},\frac{5}{2},\frac{5}{2},3]^{4}$	$[2,2,2,2,\frac{7}{2},\frac{7}{2},\frac{7}{2}]$	$[1,1,1,1,\frac{5}{2},\frac{5}{2},\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.15		$( x^{4} + x + 1 )^{4} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 )^{3} + 8 x^{3} ( x^{4} + x + 1 ) + 2$	$C_2\wr (C_2\times C_4)$ (as 16T1379)	$2048$	$2$	$[2, 2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}, 4]^{4}$	$[1,1,1,1,2,\frac{5}{2},\frac{5}{2},\frac{5}{2},3]^{4}$	$[2,2,2,2,\frac{7}{2},\frac{7}{2},\frac{7}{2}]$	$[1,1,1,1,\frac{5}{2},\frac{5}{2},\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.16		$( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x^{2} + 8 x\right) ( x^{4} + x + 1 )^{3} + 8 x^{3} ( x^{4} + x + 1 ) + 2$	$C_2\wr (C_2\times C_4)$ (as 16T1379)	$2048$	$2$	$[2, 2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}, 4]^{4}$	$[1,1,1,1,2,\frac{5}{2},\frac{5}{2},\frac{5}{2},3]^{4}$	$[2,2,2,2,\frac{7}{2},\frac{7}{2},\frac{7}{2}]$	$[1,1,1,1,\frac{5}{2},\frac{5}{2},\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.17		$( x^{4} + x + 1 )^{4} + 8 ( x^{4} + x + 1 )^{3} + 8 x^{3} ( x^{4} + x + 1 ) + 2$	$C_2\wr (C_2\times C_4)$ (as 16T1379)	$2048$	$2$	$[2, 2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}, 4]^{4}$	$[1,1,1,1,2,\frac{5}{2},\frac{5}{2},\frac{5}{2},3]^{4}$	$[2,2,2,2,\frac{7}{2},\frac{7}{2},\frac{7}{2}]$	$[1,1,1,1,\frac{5}{2},\frac{5}{2},\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.18		$( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8\right) ( x^{4} + x + 1 )^{3} + 8 x^{3} ( x^{4} + x + 1 ) + 2$	$C_2\wr (C_2\times C_4)$ (as 16T1379)	$2048$	$2$	$[2, 2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}, 4]^{4}$	$[1,1,1,1,2,\frac{5}{2},\frac{5}{2},\frac{5}{2},3]^{4}$	$[2,2,2,2,\frac{7}{2},\frac{7}{2},\frac{7}{2}]$	$[1,1,1,1,\frac{5}{2},\frac{5}{2},\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.19		$( x^{4} + x + 1 )^{4} + \left(8 x^{2} + 8\right) ( x^{4} + x + 1 )^{3} + 8 x^{3} ( x^{4} + x + 1 ) + 2$	$C_2\wr (C_2\times C_4)$ (as 16T1379)	$2048$	$2$	$[2, 2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}, 4]^{4}$	$[1,1,1,1,2,\frac{5}{2},\frac{5}{2},\frac{5}{2},3]^{4}$	$[2,2,2,2,\frac{7}{2},\frac{7}{2},\frac{7}{2}]$	$[1,1,1,1,\frac{5}{2},\frac{5}{2},\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.20		$( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x^{2} + 8\right) ( x^{4} + x + 1 )^{3} + 8 x^{3} ( x^{4} + x + 1 ) + 2$	$C_2\wr (C_2\times C_4)$ (as 16T1379)	$2048$	$2$	$[2, 2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}, 4]^{4}$	$[1,1,1,1,2,\frac{5}{2},\frac{5}{2},\frac{5}{2},3]^{4}$	$[2,2,2,2,\frac{7}{2},\frac{7}{2},\frac{7}{2}]$	$[1,1,1,1,\frac{5}{2},\frac{5}{2},\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.21		$( x^{4} + x + 1 )^{4} + \left(8 x + 8\right) ( x^{4} + x + 1 )^{3} + 8 x^{3} ( x^{4} + x + 1 ) + 2$	$C_2\wr (C_2\times C_4)$ (as 16T1379)	$2048$	$2$	$[2, 2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}, 4]^{4}$	$[1,1,1,1,2,\frac{5}{2},\frac{5}{2},\frac{5}{2},3]^{4}$	$[2,2,2,2,\frac{7}{2},\frac{7}{2},\frac{7}{2}]$	$[1,1,1,1,\frac{5}{2},\frac{5}{2},\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.22		$( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x + 8\right) ( x^{4} + x + 1 )^{3} + 8 x^{3} ( x^{4} + x + 1 ) + 2$	$C_2\wr (C_2\times C_4)$ (as 16T1379)	$2048$	$2$	$[2, 2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}, 4]^{4}$	$[1,1,1,1,2,\frac{5}{2},\frac{5}{2},\frac{5}{2},3]^{4}$	$[2,2,2,2,\frac{7}{2},\frac{7}{2},\frac{7}{2}]$	$[1,1,1,1,\frac{5}{2},\frac{5}{2},\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.23		$( x^{4} + x + 1 )^{4} + \left(8 x^{2} + 8 x + 8\right) ( x^{4} + x + 1 )^{3} + 8 x^{3} ( x^{4} + x + 1 ) + 2$	$C_2\wr (C_2\times C_4)$ (as 16T1379)	$2048$	$2$	$[2, 2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}, 4]^{4}$	$[1,1,1,1,2,\frac{5}{2},\frac{5}{2},\frac{5}{2},3]^{4}$	$[2,2,2,2,\frac{7}{2},\frac{7}{2},\frac{7}{2}]$	$[1,1,1,1,\frac{5}{2},\frac{5}{2},\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.24		$( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x^{2} + 8 x + 8\right) ( x^{4} + x + 1 )^{3} + 8 x^{3} ( x^{4} + x + 1 ) + 2$	$C_2\wr (C_2\times C_4)$ (as 16T1379)	$2048$	$2$	$[2, 2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}, 4]^{4}$	$[1,1,1,1,2,\frac{5}{2},\frac{5}{2},\frac{5}{2},3]^{4}$	$[2,2,2,2,\frac{7}{2},\frac{7}{2},\frac{7}{2}]$	$[1,1,1,1,\frac{5}{2},\frac{5}{2},\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.25		$( x^{4} + x + 1 )^{4} + 8 x ( x^{4} + x + 1 ) + 2$	$C_2^3\wr C_2:C_4$ (as 16T820)	$512$	$2$	$[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{4}$	$[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{4}$	$[2,2,2,\frac{7}{2},\frac{7}{2}]$	$[1,1,1,\frac{5}{2},\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.26		$( x^{4} + x + 1 )^{4} + 8 x ( x^{4} + x + 1 ) + 16 x^{3} + 2$	$C_2^6:(C_2\times C_4)$ (as 16T815)	$512$	$2$	$[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{4}$	$[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{4}$	$[2,2,2,\frac{7}{2},\frac{7}{2}]$	$[1,1,1,\frac{5}{2},\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.27		$( x^{4} + x + 1 )^{4} + 8 x^{3} ( x^{4} + x + 1 )^{3} + 8 x ( x^{4} + x + 1 ) + 2$	$C_2^6:(C_2\times C_4)$ (as 16T815)	$512$	$2$	$[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{4}$	$[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{4}$	$[2,2,2,\frac{7}{2},\frac{7}{2}]$	$[1,1,1,\frac{5}{2},\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.28		$( x^{4} + x + 1 )^{4} + 8 x^{3} ( x^{4} + x + 1 )^{3} + 8 x ( x^{4} + x + 1 ) + 16 x^{3} + 2$	$C_2^3\wr C_2:C_4$ (as 16T820)	$512$	$2$	$[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{4}$	$[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{4}$	$[2,2,2,\frac{7}{2},\frac{7}{2}]$	$[1,1,1,\frac{5}{2},\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.29		$( x^{4} + x + 1 )^{4} + 8 x^{2} ( x^{4} + x + 1 )^{3} + 8 x ( x^{4} + x + 1 ) + 2$	$C_2^3\wr C_2:C_4$ (as 16T820)	$512$	$2$	$[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{4}$	$[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{4}$	$[2,2,2,\frac{7}{2},\frac{7}{2}]$	$[1,1,1,\frac{5}{2},\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.30		$( x^{4} + x + 1 )^{4} + 8 x^{2} ( x^{4} + x + 1 )^{3} + 8 x ( x^{4} + x + 1 ) + 16 x^{3} + 2$	$C_2^6:(C_2\times C_4)$ (as 16T815)	$512$	$2$	$[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{4}$	$[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{4}$	$[2,2,2,\frac{7}{2},\frac{7}{2}]$	$[1,1,1,\frac{5}{2},\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.31		$( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x^{2}\right) ( x^{4} + x + 1 )^{3} + 8 x ( x^{4} + x + 1 ) + 2$	$C_2^6:(C_2\times C_4)$ (as 16T815)	$512$	$2$	$[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{4}$	$[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{4}$	$[2,2,2,\frac{7}{2},\frac{7}{2}]$	$[1,1,1,\frac{5}{2},\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.32		$( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x^{2}\right) ( x^{4} + x + 1 )^{3} + 8 x ( x^{4} + x + 1 ) + 16 x^{3} + 2$	$C_2^3\wr C_2:C_4$ (as 16T820)	$512$	$2$	$[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{4}$	$[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{4}$	$[2,2,2,\frac{7}{2},\frac{7}{2}]$	$[1,1,1,\frac{5}{2},\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.33		$( x^{4} + x + 1 )^{4} + 8 ( x^{4} + x + 1 )^{3} + 8 x ( x^{4} + x + 1 ) + 2$	$C_2^3\wr C_2:C_4$ (as 16T820)	$512$	$2$	$[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{4}$	$[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{4}$	$[2,2,2,\frac{7}{2},\frac{7}{2}]$	$[1,1,1,\frac{5}{2},\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.34		$( x^{4} + x + 1 )^{4} + 8 ( x^{4} + x + 1 )^{3} + 8 x ( x^{4} + x + 1 ) + 16 x^{3} + 2$	$C_2^6:(C_2\times C_4)$ (as 16T815)	$512$	$2$	$[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{4}$	$[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{4}$	$[2,2,2,\frac{7}{2},\frac{7}{2}]$	$[1,1,1,\frac{5}{2},\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.35		$( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8\right) ( x^{4} + x + 1 )^{3} + 8 x ( x^{4} + x + 1 ) + 2$	$C_2^6:(C_2\times C_4)$ (as 16T815)	$512$	$2$	$[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{4}$	$[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{4}$	$[2,2,2,\frac{7}{2},\frac{7}{2}]$	$[1,1,1,\frac{5}{2},\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.36		$( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8\right) ( x^{4} + x + 1 )^{3} + 8 x ( x^{4} + x + 1 ) + 16 x^{3} + 2$	$C_2^3\wr C_2:C_4$ (as 16T820)	$512$	$2$	$[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{4}$	$[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{4}$	$[2,2,2,\frac{7}{2},\frac{7}{2}]$	$[1,1,1,\frac{5}{2},\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.37		$( x^{4} + x + 1 )^{4} + \left(8 x^{2} + 8\right) ( x^{4} + x + 1 )^{3} + 8 x ( x^{4} + x + 1 ) + 2$	$C_2^3\wr C_2:C_4$ (as 16T820)	$512$	$2$	$[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{4}$	$[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{4}$	$[2,2,2,\frac{7}{2},\frac{7}{2}]$	$[1,1,1,\frac{5}{2},\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.38		$( x^{4} + x + 1 )^{4} + \left(8 x^{2} + 8\right) ( x^{4} + x + 1 )^{3} + 8 x ( x^{4} + x + 1 ) + 16 x^{3} + 2$	$C_2^6:(C_2\times C_4)$ (as 16T815)	$512$	$2$	$[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{4}$	$[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{4}$	$[2,2,2,\frac{7}{2},\frac{7}{2}]$	$[1,1,1,\frac{5}{2},\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.39		$( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x^{2} + 8\right) ( x^{4} + x + 1 )^{3} + 8 x ( x^{4} + x + 1 ) + 2$	$C_2^6:(C_2\times C_4)$ (as 16T815)	$512$	$2$	$[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{4}$	$[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{4}$	$[2,2,2,\frac{7}{2},\frac{7}{2}]$	$[1,1,1,\frac{5}{2},\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.40		$( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x^{2} + 8\right) ( x^{4} + x + 1 )^{3} + 8 x ( x^{4} + x + 1 ) + 16 x^{3} + 2$	$C_2^3\wr C_2:C_4$ (as 16T820)	$512$	$2$	$[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{4}$	$[1,1,1,2,\frac{5}{2},\frac{5}{2},3]^{4}$	$[2,2,2,\frac{7}{2},\frac{7}{2}]$	$[1,1,1,\frac{5}{2},\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.41		$( x^{4} + x + 1 )^{4} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 ) + 2$	$C_2^4.D_4$ (as 16T208)	$128$	$4$	$[2, 2, 3, \frac{7}{2}, 4]^{4}$	$[1,1,2,\frac{5}{2},3]^{4}$	$[2,2,\frac{7}{2}]$	$[1,1,\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.42		$( x^{4} + x + 1 )^{4} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 ) + 16 x^{3} + 2$	$(C_2\times C_8):D_4$ (as 16T279)	$128$	$4$	$[2, 2, 3, \frac{7}{2}, 4]^{4}$	$[1,1,2,\frac{5}{2},3]^{4}$	$[2,2,\frac{7}{2}]$	$[1,1,\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.43		$( x^{4} + x + 1 )^{4} + 8 x^{3} ( x^{4} + x + 1 )^{3} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 ) + 2$	$C_2^5:C_4$ (as 16T261)	$128$	$4$	$[2, 2, 3, \frac{7}{2}, 4]^{4}$	$[1,1,2,\frac{5}{2},3]^{4}$	$[2,2,\frac{7}{2}]$	$[1,1,\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.44		$( x^{4} + x + 1 )^{4} + 8 x^{3} ( x^{4} + x + 1 )^{3} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 ) + 16 x^{3} + 2$	$(C_2^3\times C_4):C_4$ (as 16T292)	$128$	$4$	$[2, 2, 3, \frac{7}{2}, 4]^{4}$	$[1,1,2,\frac{5}{2},3]^{4}$	$[2,2,\frac{7}{2}]$	$[1,1,\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.45		$( x^{4} + x + 1 )^{4} + 8 x ( x^{4} + x + 1 )^{3} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 ) + 2$	$C_2^4.D_4$ (as 16T268)	$128$	$2$	$[2, 2, 3, \frac{7}{2}, 4]^{4}$	$[1,1,2,\frac{5}{2},3]^{4}$	$[2,2,\frac{7}{2}]$	$[1,1,\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.46		$( x^{4} + x + 1 )^{4} + 8 x ( x^{4} + x + 1 )^{3} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 ) + 16 x^{3} + 2$	$\OD_{16}:D_4$ (as 16T278)	$128$	$2$	$[2, 2, 3, \frac{7}{2}, 4]^{4}$	$[1,1,2,\frac{5}{2},3]^{4}$	$[2,2,\frac{7}{2}]$	$[1,1,\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.47		$( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x\right) ( x^{4} + x + 1 )^{3} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 ) + 2$	$C_2^4.D_4$ (as 16T317)	$128$	$2$	$[2, 2, 3, \frac{7}{2}, 4]^{4}$	$[1,1,2,\frac{5}{2},3]^{4}$	$[2,2,\frac{7}{2}]$	$[1,1,\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.48		$( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x\right) ( x^{4} + x + 1 )^{3} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 ) + 16 x^{3} + 2$	$(C_2^2\times D_4):C_4$ (as 16T315)	$128$	$2$	$[2, 2, 3, \frac{7}{2}, 4]^{4}$	$[1,1,2,\frac{5}{2},3]^{4}$	$[2,2,\frac{7}{2}]$	$[1,1,\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.49		$( x^{4} + x + 1 )^{4} + 8 ( x^{4} + x + 1 )^{3} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 ) + 2$	$C_2^4.D_4$ (as 16T208)	$128$	$4$	$[2, 2, 3, \frac{7}{2}, 4]^{4}$	$[1,1,2,\frac{5}{2},3]^{4}$	$[2,2,\frac{7}{2}]$	$[1,1,\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.4.4.44a1.50		$( x^{4} + x + 1 )^{4} + 8 ( x^{4} + x + 1 )^{3} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 ) + 16 x^{3} + 2$	$(C_2\times C_8):D_4$ (as 16T279)	$128$	$4$	$[2, 2, 3, \frac{7}{2}, 4]^{4}$	$[1,1,2,\frac{5}{2},3]^{4}$	$[2,2,\frac{7}{2}]$	$[1,1,\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$

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