LMFDB - $p$-adic family 2.1.16.42b

Defining polynomial

$x^{16} + 4 b_{31} x^{15} + 2 a_{14} x^{14} + 4 b_{29} x^{13} + 2 b_{12} x^{12} + 4 a_{27} x^{11} + \left(2 b_{8} + 8 c_{40}\right) x^{8} + 8 b_{39} x^{7} + 8 b_{37} x^{5} + 8 b_{35} x^{3} + 8 b_{33} x + 4 c_{16} + 2$

Invariants

Residue field characteristic:	$2$
Degree:	$16$
Base field:	$\Q_{2}$
Ramification index $e$:	$16$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$42$
Artin slopes:	$[2,2,2,\frac{7}{2}]$
Swan slopes:	$[1,1,1,\frac{5}{2}]$
Means:	$\langle\frac{1}{2},\frac{3}{4},\frac{7}{8},\frac{27}{16}\rangle$
Rams:	$(1,1,1,13)$
Field count:	$544$ (complete)
Ambiguity:	$4$
Mass:	$256$
Absolute Mass:	$256$

Diagrams

Varying

Indices of inseparability:	$[27,14,12,8,0]$ (show 144), $[27,14,12,12,0]$ (show 128), $[27,14,14,8,0]$ (show 128), $[27,14,14,14,0]$ (show 144)
Associated inertia:	$[3,1]$ (show 144), $[4,1]$ (show 144), $[7,1]$ (show 256)
Jump Set:	$[1,2,4,15,31]$ (show 128), $[1,2,7,14,32]$ (show 72), $[1,2,7,23,39]$ (show 72), $[1,3,6,15,31]$ (show 128), $[1,3,7,14,32]$ (show 72), $[1,3,7,23,39]$ (show 72)

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 96
		$D_4\times A_4$ (as 16T179)	$D_4.A_4$ (as 16T180)	Total
hidden slopes	$[3]^{3}$	8	8	16

		Galois groups of order 128
		$C_2^4.D_4$ (as 16T230)	$C_2^4.D_4$ (as 16T254)	$C_2^4.D_4$ (as 16T297)	$C_2^4.D_4$ (as 16T319)	$C_4^2.D_4$ (as 16T361)	$C_4^2.D_4$ (as 16T387)	$C_4^2.D_4$ (as 16T397)	$C_4^2.D_4$ (as 16T398)	Total
hidden slopes	$[3]^{4}$	8	8	4	4	4	4	8	8	48

		Galois groups of order 192
		$(C_2^2\times C_4):A_4$ (as 16T426)	$C_2\wr A_4$ (as 16T427)	Total
hidden slopes	$[3,3]^{3}$	8	8	16

		Galois groups of order 384
		$C_2^4:(C_2\times A_4)$ (as 16T716)	$(C_2^3\times C_4):A_4$ (as 16T717)	$(C_2^2\times D_4):A_4$ (as 16T720)	$C_2^4:(C_2\times A_4)$ (as 16T721)	$C_2^5:A_4$ (as 16T722)	Total
hidden slopes	$[3,3]^{6}$	8	8	4	4	8	32

		Galois groups of order 512
		$C_2^5.(C_2\times D_4)$ (as 16T874)	$D_4:C_2^3.D_4$ (as 16T918)	Total
hidden slopes	$[3,3,3]^{4}$	16	16	32

		Galois groups of order 768
		$(D_4\times C_2^3):A_4$ (as 16T1037)
hidden slopes	$[2,2,3]^{6}$	16

		Galois groups of order 896
		$C_2^4.F_8$ (as 16T1076)
hidden slopes	$[3,3,3]^{7}$	128

		Galois groups of order 1024
		$(D_4\times C_2^3).Q_{16}$ (as 16T1255)	$C_2^6.\SD_{16}$ (as 16T1284)	Total
hidden slopes	$[2,3,3,3]_{4}$	16	16	32

$[2,3,\frac{5}{2},\frac{5}{2}]_{4}$	16	16	32

Total	32	32	64

		Galois groups of order 6144
		$C_2\wr C_2^3:C_3$ (as 16T1658)
hidden slopes	$[2,2,2,\frac{5}{2},3,3]^{6}$	24

$[2,2,2,\frac{5}{2},\frac{5}{2},3]^{6}$	32

Total	56

Fields

Galois group	Galois Artin	Indices

Assoc. Inertia	Jump Set

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Showing 1-50 of 128

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Label	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
2.1.16.42b3.1	$x^{16} + 2 x^{14} + 4 x^{11} + 2 x^{8} + 2$	$C_2^4.F_8$ (as 16T1076)	$896$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{7}$	$[1,1,1,2,2,2,\frac{5}{2}]^{7}$	$[3,3,3]^{7}$	$[2,2,2]^{7}$	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.2	$x^{16} + 2 x^{14} + 4 x^{11} + 10 x^{8} + 2$	$C_2^4.F_8$ (as 16T1076)	$896$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{7}$	$[1,1,1,2,2,2,\frac{5}{2}]^{7}$	$[3,3,3]^{7}$	$[2,2,2]^{7}$	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.3	$x^{16} + 2 x^{14} + 4 x^{11} + 2 x^{8} + 8 x^{7} + 2$	$C_2^4.F_8$ (as 16T1076)	$896$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{7}$	$[1,1,1,2,2,2,\frac{5}{2}]^{7}$	$[3,3,3]^{7}$	$[2,2,2]^{7}$	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.4	$x^{16} + 2 x^{14} + 4 x^{11} + 10 x^{8} + 8 x^{7} + 2$	$C_2^4.F_8$ (as 16T1076)	$896$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{7}$	$[1,1,1,2,2,2,\frac{5}{2}]^{7}$	$[3,3,3]^{7}$	$[2,2,2]^{7}$	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.5	$x^{16} + 2 x^{14} + 4 x^{11} + 2 x^{8} + 8 x^{5} + 2$	$C_2^4.F_8$ (as 16T1076)	$896$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{7}$	$[1,1,1,2,2,2,\frac{5}{2}]^{7}$	$[3,3,3]^{7}$	$[2,2,2]^{7}$	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.6	$x^{16} + 2 x^{14} + 4 x^{11} + 10 x^{8} + 8 x^{5} + 2$	$C_2^4.F_8$ (as 16T1076)	$896$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{7}$	$[1,1,1,2,2,2,\frac{5}{2}]^{7}$	$[3,3,3]^{7}$	$[2,2,2]^{7}$	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.7	$x^{16} + 2 x^{14} + 4 x^{11} + 2 x^{8} + 8 x^{7} + 8 x^{5} + 2$	$C_2^4.F_8$ (as 16T1076)	$896$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{7}$	$[1,1,1,2,2,2,\frac{5}{2}]^{7}$	$[3,3,3]^{7}$	$[2,2,2]^{7}$	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.8	$x^{16} + 2 x^{14} + 4 x^{11} + 10 x^{8} + 8 x^{7} + 8 x^{5} + 2$	$C_2^4.F_8$ (as 16T1076)	$896$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{7}$	$[1,1,1,2,2,2,\frac{5}{2}]^{7}$	$[3,3,3]^{7}$	$[2,2,2]^{7}$	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.9	$x^{16} + 2 x^{14} + 4 x^{11} + 2 x^{8} + 8 x^{3} + 2$	$C_2^4.F_8$ (as 16T1076)	$896$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{7}$	$[1,1,1,2,2,2,\frac{5}{2}]^{7}$	$[3,3,3]^{7}$	$[2,2,2]^{7}$	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.10	$x^{16} + 2 x^{14} + 4 x^{11} + 10 x^{8} + 8 x^{3} + 2$	$C_2^4.F_8$ (as 16T1076)	$896$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{7}$	$[1,1,1,2,2,2,\frac{5}{2}]^{7}$	$[3,3,3]^{7}$	$[2,2,2]^{7}$	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.11	$x^{16} + 2 x^{14} + 4 x^{11} + 2 x^{8} + 8 x^{7} + 8 x^{3} + 2$	$C_2^4.F_8$ (as 16T1076)	$896$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{7}$	$[1,1,1,2,2,2,\frac{5}{2}]^{7}$	$[3,3,3]^{7}$	$[2,2,2]^{7}$	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.12	$x^{16} + 2 x^{14} + 4 x^{11} + 10 x^{8} + 8 x^{7} + 8 x^{3} + 2$	$C_2^4.F_8$ (as 16T1076)	$896$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{7}$	$[1,1,1,2,2,2,\frac{5}{2}]^{7}$	$[3,3,3]^{7}$	$[2,2,2]^{7}$	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.13	$x^{16} + 2 x^{14} + 4 x^{11} + 2 x^{8} + 8 x^{5} + 8 x^{3} + 2$	$C_2^4.F_8$ (as 16T1076)	$896$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{7}$	$[1,1,1,2,2,2,\frac{5}{2}]^{7}$	$[3,3,3]^{7}$	$[2,2,2]^{7}$	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.14	$x^{16} + 2 x^{14} + 4 x^{11} + 10 x^{8} + 8 x^{5} + 8 x^{3} + 2$	$C_2^4.F_8$ (as 16T1076)	$896$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{7}$	$[1,1,1,2,2,2,\frac{5}{2}]^{7}$	$[3,3,3]^{7}$	$[2,2,2]^{7}$	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.15	$x^{16} + 2 x^{14} + 4 x^{11} + 2 x^{8} + 8 x^{7} + 8 x^{5} + 8 x^{3} + 2$	$C_2^4.F_8$ (as 16T1076)	$896$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{7}$	$[1,1,1,2,2,2,\frac{5}{2}]^{7}$	$[3,3,3]^{7}$	$[2,2,2]^{7}$	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.16	$x^{16} + 2 x^{14} + 4 x^{11} + 10 x^{8} + 8 x^{7} + 8 x^{5} + 8 x^{3} + 2$	$C_2^4.F_8$ (as 16T1076)	$896$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{7}$	$[1,1,1,2,2,2,\frac{5}{2}]^{7}$	$[3,3,3]^{7}$	$[2,2,2]^{7}$	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.17	$x^{16} + 2 x^{14} + 4 x^{11} + 2 x^{8} + 8 x + 2$	$C_2^8.F_8$ (as 16T1768)	$14336$	$2$	not computed	not computed	not computed	not computed	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.18	$x^{16} + 2 x^{14} + 4 x^{11} + 10 x^{8} + 8 x + 2$	$C_2^8.F_8$ (as 16T1768)	$14336$	$2$	not computed	not computed	not computed	not computed	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.19	$x^{16} + 2 x^{14} + 4 x^{11} + 2 x^{8} + 8 x^{7} + 8 x + 2$	$C_2^8.F_8$ (as 16T1768)	$14336$	$2$	not computed	not computed	not computed	not computed	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.20	$x^{16} + 2 x^{14} + 4 x^{11} + 10 x^{8} + 8 x^{7} + 8 x + 2$	$C_2^8.F_8$ (as 16T1768)	$14336$	$2$	not computed	not computed	not computed	not computed	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.21	$x^{16} + 2 x^{14} + 4 x^{11} + 2 x^{8} + 8 x^{5} + 8 x + 2$	$C_2^8.F_8$ (as 16T1768)	$14336$	$2$	not computed	not computed	not computed	not computed	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.22	$x^{16} + 2 x^{14} + 4 x^{11} + 10 x^{8} + 8 x^{5} + 8 x + 2$	$C_2^8.F_8$ (as 16T1768)	$14336$	$2$	not computed	not computed	not computed	not computed	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.23	$x^{16} + 2 x^{14} + 4 x^{11} + 2 x^{8} + 8 x^{7} + 8 x^{5} + 8 x + 2$	$C_2^8.F_8$ (as 16T1768)	$14336$	$2$	not computed	not computed	not computed	not computed	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.24	$x^{16} + 2 x^{14} + 4 x^{11} + 10 x^{8} + 8 x^{7} + 8 x^{5} + 8 x + 2$	$C_2^8.F_8$ (as 16T1768)	$14336$	$2$	not computed	not computed	not computed	not computed	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.25	$x^{16} + 2 x^{14} + 4 x^{11} + 2 x^{8} + 8 x^{3} + 8 x + 2$	$C_2^8.F_8$ (as 16T1768)	$14336$	$2$	not computed	not computed	not computed	not computed	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.26	$x^{16} + 2 x^{14} + 4 x^{11} + 10 x^{8} + 8 x^{3} + 8 x + 2$	$C_2^8.F_8$ (as 16T1768)	$14336$	$2$	not computed	not computed	not computed	not computed	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.27	$x^{16} + 2 x^{14} + 4 x^{11} + 2 x^{8} + 8 x^{7} + 8 x^{3} + 8 x + 2$	$C_2^8.F_8$ (as 16T1768)	$14336$	$2$	not computed	not computed	not computed	not computed	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.28	$x^{16} + 2 x^{14} + 4 x^{11} + 10 x^{8} + 8 x^{7} + 8 x^{3} + 8 x + 2$	$C_2^8.F_8$ (as 16T1768)	$14336$	$2$	not computed	not computed	not computed	not computed	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.29	$x^{16} + 2 x^{14} + 4 x^{11} + 2 x^{8} + 8 x^{5} + 8 x^{3} + 8 x + 2$	$C_2^8.F_8$ (as 16T1768)	$14336$	$2$	not computed	not computed	not computed	not computed	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.30	$x^{16} + 2 x^{14} + 4 x^{11} + 10 x^{8} + 8 x^{5} + 8 x^{3} + 8 x + 2$	$C_2^8.F_8$ (as 16T1768)	$14336$	$2$	not computed	not computed	not computed	not computed	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.31	$x^{16} + 2 x^{14} + 4 x^{11} + 2 x^{8} + 8 x^{7} + 8 x^{5} + 8 x^{3} + 8 x + 2$	$C_2^8.F_8$ (as 16T1768)	$14336$	$2$	not computed	not computed	not computed	not computed	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.32	$x^{16} + 2 x^{14} + 4 x^{11} + 10 x^{8} + 8 x^{7} + 8 x^{5} + 8 x^{3} + 8 x + 2$	$C_2^8.F_8$ (as 16T1768)	$14336$	$2$	not computed	not computed	not computed	not computed	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.33	$x^{16} + 4 x^{15} + 2 x^{14} + 4 x^{11} + 2 x^{8} + 2$	$C_2^4.F_8$ (as 16T1076)	$896$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{7}$	$[1,1,1,2,2,2,\frac{5}{2}]^{7}$	$[3,3,3]^{7}$	$[2,2,2]^{7}$	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.34	$x^{16} + 4 x^{15} + 2 x^{14} + 4 x^{11} + 10 x^{8} + 2$	$C_2^4.F_8$ (as 16T1076)	$896$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{7}$	$[1,1,1,2,2,2,\frac{5}{2}]^{7}$	$[3,3,3]^{7}$	$[2,2,2]^{7}$	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.35	$x^{16} + 4 x^{15} + 2 x^{14} + 4 x^{11} + 2 x^{8} + 8 x^{7} + 2$	$C_2^4.F_8$ (as 16T1076)	$896$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{7}$	$[1,1,1,2,2,2,\frac{5}{2}]^{7}$	$[3,3,3]^{7}$	$[2,2,2]^{7}$	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.36	$x^{16} + 4 x^{15} + 2 x^{14} + 4 x^{11} + 10 x^{8} + 8 x^{7} + 2$	$C_2^4.F_8$ (as 16T1076)	$896$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{7}$	$[1,1,1,2,2,2,\frac{5}{2}]^{7}$	$[3,3,3]^{7}$	$[2,2,2]^{7}$	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.37	$x^{16} + 4 x^{15} + 2 x^{14} + 4 x^{11} + 2 x^{8} + 8 x^{5} + 2$	$C_2^4.F_8$ (as 16T1076)	$896$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{7}$	$[1,1,1,2,2,2,\frac{5}{2}]^{7}$	$[3,3,3]^{7}$	$[2,2,2]^{7}$	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.38	$x^{16} + 4 x^{15} + 2 x^{14} + 4 x^{11} + 10 x^{8} + 8 x^{5} + 2$	$C_2^4.F_8$ (as 16T1076)	$896$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{7}$	$[1,1,1,2,2,2,\frac{5}{2}]^{7}$	$[3,3,3]^{7}$	$[2,2,2]^{7}$	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.39	$x^{16} + 4 x^{15} + 2 x^{14} + 4 x^{11} + 2 x^{8} + 8 x^{7} + 8 x^{5} + 2$	$C_2^4.F_8$ (as 16T1076)	$896$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{7}$	$[1,1,1,2,2,2,\frac{5}{2}]^{7}$	$[3,3,3]^{7}$	$[2,2,2]^{7}$	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.40	$x^{16} + 4 x^{15} + 2 x^{14} + 4 x^{11} + 10 x^{8} + 8 x^{7} + 8 x^{5} + 2$	$C_2^4.F_8$ (as 16T1076)	$896$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{7}$	$[1,1,1,2,2,2,\frac{5}{2}]^{7}$	$[3,3,3]^{7}$	$[2,2,2]^{7}$	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.41	$x^{16} + 4 x^{15} + 2 x^{14} + 4 x^{11} + 2 x^{8} + 8 x^{3} + 2$	$C_2^4.F_8$ (as 16T1076)	$896$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{7}$	$[1,1,1,2,2,2,\frac{5}{2}]^{7}$	$[3,3,3]^{7}$	$[2,2,2]^{7}$	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.42	$x^{16} + 4 x^{15} + 2 x^{14} + 4 x^{11} + 10 x^{8} + 8 x^{3} + 2$	$C_2^4.F_8$ (as 16T1076)	$896$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{7}$	$[1,1,1,2,2,2,\frac{5}{2}]^{7}$	$[3,3,3]^{7}$	$[2,2,2]^{7}$	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.43	$x^{16} + 4 x^{15} + 2 x^{14} + 4 x^{11} + 2 x^{8} + 8 x^{7} + 8 x^{3} + 2$	$C_2^4.F_8$ (as 16T1076)	$896$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{7}$	$[1,1,1,2,2,2,\frac{5}{2}]^{7}$	$[3,3,3]^{7}$	$[2,2,2]^{7}$	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.44	$x^{16} + 4 x^{15} + 2 x^{14} + 4 x^{11} + 10 x^{8} + 8 x^{7} + 8 x^{3} + 2$	$C_2^4.F_8$ (as 16T1076)	$896$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{7}$	$[1,1,1,2,2,2,\frac{5}{2}]^{7}$	$[3,3,3]^{7}$	$[2,2,2]^{7}$	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.45	$x^{16} + 4 x^{15} + 2 x^{14} + 4 x^{11} + 2 x^{8} + 8 x^{5} + 8 x^{3} + 2$	$C_2^4.F_8$ (as 16T1076)	$896$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{7}$	$[1,1,1,2,2,2,\frac{5}{2}]^{7}$	$[3,3,3]^{7}$	$[2,2,2]^{7}$	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.46	$x^{16} + 4 x^{15} + 2 x^{14} + 4 x^{11} + 10 x^{8} + 8 x^{5} + 8 x^{3} + 2$	$C_2^4.F_8$ (as 16T1076)	$896$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{7}$	$[1,1,1,2,2,2,\frac{5}{2}]^{7}$	$[3,3,3]^{7}$	$[2,2,2]^{7}$	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.47	$x^{16} + 4 x^{15} + 2 x^{14} + 4 x^{11} + 2 x^{8} + 8 x^{7} + 8 x^{5} + 8 x^{3} + 2$	$C_2^4.F_8$ (as 16T1076)	$896$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{7}$	$[1,1,1,2,2,2,\frac{5}{2}]^{7}$	$[3,3,3]^{7}$	$[2,2,2]^{7}$	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.48	$x^{16} + 4 x^{15} + 2 x^{14} + 4 x^{11} + 10 x^{8} + 8 x^{7} + 8 x^{5} + 8 x^{3} + 2$	$C_2^4.F_8$ (as 16T1076)	$896$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{7}$	$[1,1,1,2,2,2,\frac{5}{2}]^{7}$	$[3,3,3]^{7}$	$[2,2,2]^{7}$	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.49	$x^{16} + 4 x^{15} + 2 x^{14} + 4 x^{11} + 2 x^{8} + 8 x + 2$	$C_2^8.F_8$ (as 16T1768)	$14336$	$2$	not computed	not computed	not computed	not computed	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$
2.1.16.42b3.50	$x^{16} + 4 x^{15} + 2 x^{14} + 4 x^{11} + 10 x^{8} + 8 x + 2$	$C_2^8.F_8$ (as 16T1768)	$14336$	$2$	not computed	not computed	not computed	not computed	$[27, 14, 14, 8, 0]$	$[7, 1]$	$z^{14} + z^6 + 1,z + 1$	$[1, 2, 4, 15, 31]$

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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
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Belyi maps

Label	2.1.16.42b
Base	2.1.1.0a1.1
Degree	\(16\)
e	\(16\)
f	\(1\)
c	\(42\)

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