LMFDB - $p$-adic family 2.1.16.40c

Defining polynomial

$x^{16} + 4 b_{31} x^{15} + 2 a_{14} x^{14} + 4 b_{29} x^{13} + 2 b_{12} x^{12} + 4 b_{27} x^{11} + 4 a_{25} x^{9} + 2 b_{8} x^{8} + 8 c_{36} x^{4} + 8 b_{35} x^{3} + 8 b_{33} x + 4 c_{16} + 2$

Invariants

Residue field characteristic:	$2$
Degree:	$16$
Base field:	$\Q_{2}$
Ramification index $e$:	$16$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$40$
Artin slopes:	$[2,2,2,\frac{13}{4}]$
Swan slopes:	$[1,1,1,\frac{9}{4}]$
Means:	$\langle\frac{1}{2},\frac{3}{4},\frac{7}{8},\frac{25}{16}\rangle$
Rams:	$(1,1,1,11)$
Field count:	$256$ (complete)
Ambiguity:	$4$
Mass:	$128$
Absolute Mass:	$128$

Diagrams

Varying

Indices of inseparability:	$[25,14,12,8,0]$ (show 64), $[25,14,12,12,0]$ (show 64), $[25,14,14,8,0]$ (show 64), $[25,14,14,14,0]$ (show 64)
Associated inertia:	$[3,1]$ (show 64), $[4,1]$ (show 64), $[7,1]$ (show 128)
Jump Set:	$[1,2,4,15,31]$ (show 64), $[1,2,7,14,32]$ (show 32), $[1,2,7,23,39]$ (show 32), $[1,3,6,15,31]$ (show 64), $[1,3,7,14,32]$ (show 32), $[1,3,7,23,39]$ (show 32)

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 384
		$C_2^4:(C_2\times A_4)$ (as 16T716)
hidden slopes	$[3,3,3]^{3}$	16

		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	$[3,3,3]^{6}$	16

		Galois groups of order 896
		$C_2^4.F_8$ (as 16T1076)
hidden slopes	$[3,3,3]^{7}$	64

		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	$[2,3,3,3]^{4}$	12	14	26

$[2,3,3,3]_{4}$	4	2	6

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

		Galois groups of order 14336
		$C_2^8.F_8$ (as 16T1768)
hidden slopes	$[2,2,2,3,3,3]^{14}$	64

Fields

Galois group	Galois Artin	Indices

Assoc. Inertia	Jump Set

Sort order	Select

Showing all 32

Download displayed columns for results to

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
2.1.16.40c1.33	$x^{16} + 2 x^{14} + 4 x^{9} + 6$	$(D_4\times C_2^3):A_4$ (as 16T1037)	$768$	$2$	$[2, 2, 2, 3, 3, 3, \frac{13}{4}]^{6}$	$[1,1,1,2,2,2,\frac{9}{4}]^{6}$	$[3,3,3]^{6}$	$[2,2,2]^{6}$	$[25, 14, 14, 14, 0]$	$[3, 1]$	$z^{14} + 1,z + 1$	$[1, 3, 7, 14, 32]$
2.1.16.40c1.34	$x^{16} + 2 x^{14} + 4 x^{9} + 8 x^{4} + 6$	$(D_4\times C_2^3):A_4$ (as 16T1037)	$768$	$2$	$[2, 2, 2, 3, 3, 3, \frac{13}{4}]^{6}$	$[1,1,1,2,2,2,\frac{9}{4}]^{6}$	$[3,3,3]^{6}$	$[2,2,2]^{6}$	$[25, 14, 14, 14, 0]$	$[3, 1]$	$z^{14} + 1,z + 1$	$[1, 3, 7, 14, 32]$
2.1.16.40c1.35	$x^{16} + 2 x^{14} + 4 x^{9} + 8 x^{3} + 6$	$(D_4\times C_2^3):A_4$ (as 16T1037)	$768$	$2$	$[2, 2, 2, 3, 3, 3, \frac{13}{4}]^{6}$	$[1,1,1,2,2,2,\frac{9}{4}]^{6}$	$[3,3,3]^{6}$	$[2,2,2]^{6}$	$[25, 14, 14, 14, 0]$	$[3, 1]$	$z^{14} + 1,z + 1$	$[1, 3, 7, 14, 32]$
2.1.16.40c1.36	$x^{16} + 2 x^{14} + 4 x^{9} + 8 x^{4} + 8 x^{3} + 6$	$(D_4\times C_2^3):A_4$ (as 16T1037)	$768$	$2$	$[2, 2, 2, 3, 3, 3, \frac{13}{4}]^{6}$	$[1,1,1,2,2,2,\frac{9}{4}]^{6}$	$[3,3,3]^{6}$	$[2,2,2]^{6}$	$[25, 14, 14, 14, 0]$	$[3, 1]$	$z^{14} + 1,z + 1$	$[1, 3, 7, 14, 32]$
2.1.16.40c1.37	$x^{16} + 4 x^{15} + 2 x^{14} + 4 x^{9} + 6$	$C_2^4:(C_2\times A_4)$ (as 16T716)	$384$	$2$	$[2, 2, 2, 3, 3, 3, \frac{13}{4}]^{3}$	$[1,1,1,2,2,2,\frac{9}{4}]^{3}$	$[3,3,3]^{3}$	$[2,2,2]^{3}$	$[25, 14, 14, 14, 0]$	$[3, 1]$	$z^{14} + 1,z + 1$	$[1, 3, 7, 14, 32]$
2.1.16.40c1.38	$x^{16} + 4 x^{15} + 2 x^{14} + 4 x^{9} + 8 x^{4} + 6$	$C_2^4:(C_2\times A_4)$ (as 16T716)	$384$	$2$	$[2, 2, 2, 3, 3, 3, \frac{13}{4}]^{3}$	$[1,1,1,2,2,2,\frac{9}{4}]^{3}$	$[3,3,3]^{3}$	$[2,2,2]^{3}$	$[25, 14, 14, 14, 0]$	$[3, 1]$	$z^{14} + 1,z + 1$	$[1, 3, 7, 14, 32]$
2.1.16.40c1.39	$x^{16} + 4 x^{15} + 2 x^{14} + 4 x^{9} + 8 x^{3} + 6$	$C_2^4:(C_2\times A_4)$ (as 16T716)	$384$	$2$	$[2, 2, 2, 3, 3, 3, \frac{13}{4}]^{3}$	$[1,1,1,2,2,2,\frac{9}{4}]^{3}$	$[3,3,3]^{3}$	$[2,2,2]^{3}$	$[25, 14, 14, 14, 0]$	$[3, 1]$	$z^{14} + 1,z + 1$	$[1, 3, 7, 14, 32]$
2.1.16.40c1.40	$x^{16} + 4 x^{15} + 2 x^{14} + 4 x^{9} + 8 x^{4} + 8 x^{3} + 6$	$C_2^4:(C_2\times A_4)$ (as 16T716)	$384$	$2$	$[2, 2, 2, 3, 3, 3, \frac{13}{4}]^{3}$	$[1,1,1,2,2,2,\frac{9}{4}]^{3}$	$[3,3,3]^{3}$	$[2,2,2]^{3}$	$[25, 14, 14, 14, 0]$	$[3, 1]$	$z^{14} + 1,z + 1$	$[1, 3, 7, 14, 32]$
2.1.16.40c1.41	$x^{16} + 2 x^{14} + 4 x^{13} + 4 x^{9} + 6$	$C_2\wr C_2^3:C_3$ (as 16T1658)	$6144$	$2$	not computed	not computed	not computed	not computed	$[25, 14, 14, 14, 0]$	$[3, 1]$	$z^{14} + 1,z + 1$	$[1, 3, 7, 14, 32]$
2.1.16.40c1.42	$x^{16} + 2 x^{14} + 4 x^{13} + 4 x^{9} + 8 x^{4} + 6$	$C_2\wr C_2^3:C_3$ (as 16T1658)	$6144$	$2$	not computed	not computed	not computed	not computed	$[25, 14, 14, 14, 0]$	$[3, 1]$	$z^{14} + 1,z + 1$	$[1, 3, 7, 14, 32]$
2.1.16.40c1.43	$x^{16} + 2 x^{14} + 4 x^{13} + 4 x^{9} + 8 x^{3} + 6$	$C_2\wr C_2^3:C_3$ (as 16T1658)	$6144$	$2$	not computed	not computed	not computed	not computed	$[25, 14, 14, 14, 0]$	$[3, 1]$	$z^{14} + 1,z + 1$	$[1, 3, 7, 14, 32]$
2.1.16.40c1.44	$x^{16} + 2 x^{14} + 4 x^{13} + 4 x^{9} + 8 x + 6$	$C_2\wr C_2^3:C_3$ (as 16T1658)	$6144$	$2$	$[2, 2, 2, 2, 2, 2, 3, 3, 3, \frac{13}{4}]^{6}$	$[1,1,1,1,1,1,2,2,2,\frac{9}{4}]^{6}$	$[2,2,2,3,3,3]^{6}$	$[1,1,1,2,2,2]^{6}$	$[25, 14, 14, 14, 0]$	$[3, 1]$	$z^{14} + 1,z + 1$	$[1, 3, 7, 14, 32]$
2.1.16.40c1.45	$x^{16} + 2 x^{14} + 4 x^{13} + 4 x^{9} + 8 x^{3} + 8 x + 6$	$C_2\wr C_2^3:C_3$ (as 16T1658)	$6144$	$2$	$[2, 2, 2, 2, 2, 2, 3, 3, 3, \frac{13}{4}]^{6}$	$[1,1,1,1,1,1,2,2,2,\frac{9}{4}]^{6}$	$[2,2,2,3,3,3]^{6}$	$[1,1,1,2,2,2]^{6}$	$[25, 14, 14, 14, 0]$	$[3, 1]$	$z^{14} + 1,z + 1$	$[1, 3, 7, 14, 32]$
2.1.16.40c1.46	$x^{16} + 2 x^{14} + 4 x^{13} + 4 x^{9} + 8 x^{4} + 8 x^{3} + 8 x + 6$	$C_2\wr C_2^3:C_3$ (as 16T1658)	$6144$	$2$	$[2, 2, 2, 2, 2, 2, 3, 3, 3, \frac{13}{4}]^{6}$	$[1,1,1,1,1,1,2,2,2,\frac{9}{4}]^{6}$	$[2,2,2,3,3,3]^{6}$	$[1,1,1,2,2,2]^{6}$	$[25, 14, 14, 14, 0]$	$[3, 1]$	$z^{14} + 1,z + 1$	$[1, 3, 7, 14, 32]$
2.1.16.40c1.47	$x^{16} + 4 x^{15} + 2 x^{14} + 4 x^{13} + 4 x^{9} + 6$	$C_2\wr C_2^3:C_3$ (as 16T1658)	$6144$	$2$	not computed	not computed	not computed	not computed	$[25, 14, 14, 14, 0]$	$[3, 1]$	$z^{14} + 1,z + 1$	$[1, 3, 7, 14, 32]$
2.1.16.40c1.48	$x^{16} + 4 x^{15} + 2 x^{14} + 4 x^{13} + 4 x^{9} + 8 x^{4} + 6$	$C_2\wr C_2^3:C_3$ (as 16T1658)	$6144$	$2$	not computed	not computed	not computed	not computed	$[25, 14, 14, 14, 0]$	$[3, 1]$	$z^{14} + 1,z + 1$	$[1, 3, 7, 14, 32]$
2.1.16.40c1.49	$x^{16} + 4 x^{15} + 2 x^{14} + 4 x^{13} + 4 x^{9} + 8 x + 6$	$C_2\wr C_2^3:C_3$ (as 16T1658)	$6144$	$2$	$[2, 2, 2, 2, 2, 2, 3, 3, 3, \frac{13}{4}]^{6}$	$[1,1,1,1,1,1,2,2,2,\frac{9}{4}]^{6}$	$[2,2,2,3,3,3]^{6}$	$[1,1,1,2,2,2]^{6}$	$[25, 14, 14, 14, 0]$	$[3, 1]$	$z^{14} + 1,z + 1$	$[1, 3, 7, 14, 32]$
2.1.16.40c1.50	$x^{16} + 4 x^{15} + 2 x^{14} + 4 x^{13} + 4 x^{9} + 8 x^{4} + 8 x + 6$	$C_2\wr C_2^3:C_3$ (as 16T1658)	$6144$	$2$	$[2, 2, 2, 2, 2, 2, 3, 3, 3, \frac{13}{4}]^{6}$	$[1,1,1,1,1,1,2,2,2,\frac{9}{4}]^{6}$	$[2,2,2,3,3,3]^{6}$	$[1,1,1,2,2,2]^{6}$	$[25, 14, 14, 14, 0]$	$[3, 1]$	$z^{14} + 1,z + 1$	$[1, 3, 7, 14, 32]$
2.1.16.40c1.51	$x^{16} + 2 x^{14} + 4 x^{11} + 4 x^{9} + 8 x^{3} + 6$	$C_2\wr C_2^3:C_3$ (as 16T1658)	$6144$	$2$	not computed	not computed	not computed	not computed	$[25, 14, 14, 14, 0]$	$[3, 1]$	$z^{14} + 1,z + 1$	$[1, 3, 7, 14, 32]$
2.1.16.40c1.52	$x^{16} + 2 x^{14} + 4 x^{11} + 4 x^{9} + 8 x + 6$	$C_2\wr C_2^3:C_3$ (as 16T1658)	$6144$	$2$	$[2, 2, 2, 2, 2, 2, 3, 3, 3, \frac{13}{4}]^{6}$	$[1,1,1,1,1,1,2,2,2,\frac{9}{4}]^{6}$	$[2,2,2,3,3,3]^{6}$	$[1,1,1,2,2,2]^{6}$	$[25, 14, 14, 14, 0]$	$[3, 1]$	$z^{14} + 1,z + 1$	$[1, 3, 7, 14, 32]$
2.1.16.40c1.53	$x^{16} + 4 x^{15} + 2 x^{14} + 4 x^{11} + 4 x^{9} + 6$	$C_2\wr C_2^3:C_3$ (as 16T1658)	$6144$	$2$	not computed	not computed	not computed	not computed	$[25, 14, 14, 14, 0]$	$[3, 1]$	$z^{14} + 1,z + 1$	$[1, 3, 7, 14, 32]$
2.1.16.40c1.54	$x^{16} + 4 x^{15} + 2 x^{14} + 4 x^{11} + 4 x^{9} + 8 x^{4} + 6$	$C_2\wr C_2^3:C_3$ (as 16T1658)	$6144$	$2$	not computed	not computed	not computed	not computed	$[25, 14, 14, 14, 0]$	$[3, 1]$	$z^{14} + 1,z + 1$	$[1, 3, 7, 14, 32]$
2.1.16.40c1.55	$x^{16} + 4 x^{15} + 2 x^{14} + 4 x^{11} + 4 x^{9} + 8 x + 6$	$C_2\wr C_2^3:C_3$ (as 16T1658)	$6144$	$2$	$[2, 2, 2, 2, 2, 2, 3, 3, 3, \frac{13}{4}]^{6}$	$[1,1,1,1,1,1,2,2,2,\frac{9}{4}]^{6}$	$[2,2,2,3,3,3]^{6}$	$[1,1,1,2,2,2]^{6}$	$[25, 14, 14, 14, 0]$	$[3, 1]$	$z^{14} + 1,z + 1$	$[1, 3, 7, 14, 32]$
2.1.16.40c1.56	$x^{16} + 4 x^{15} + 2 x^{14} + 4 x^{11} + 4 x^{9} + 8 x^{4} + 8 x + 6$	$C_2\wr C_2^3:C_3$ (as 16T1658)	$6144$	$2$	$[2, 2, 2, 2, 2, 2, 3, 3, 3, \frac{13}{4}]^{6}$	$[1,1,1,1,1,1,2,2,2,\frac{9}{4}]^{6}$	$[2,2,2,3,3,3]^{6}$	$[1,1,1,2,2,2]^{6}$	$[25, 14, 14, 14, 0]$	$[3, 1]$	$z^{14} + 1,z + 1$	$[1, 3, 7, 14, 32]$
2.1.16.40c1.57	$x^{16} + 2 x^{14} + 4 x^{13} + 4 x^{11} + 4 x^{9} + 6$	$(D_4\times C_2^3):A_4$ (as 16T1037)	$768$	$2$	$[2, 2, 2, 3, 3, 3, \frac{13}{4}]^{6}$	$[1,1,1,2,2,2,\frac{9}{4}]^{6}$	$[3,3,3]^{6}$	$[2,2,2]^{6}$	$[25, 14, 14, 14, 0]$	$[3, 1]$	$z^{14} + 1,z + 1$	$[1, 3, 7, 14, 32]$
2.1.16.40c1.58	$x^{16} + 2 x^{14} + 4 x^{13} + 4 x^{11} + 4 x^{9} + 8 x^{4} + 6$	$(D_4\times C_2^3):A_4$ (as 16T1037)	$768$	$2$	$[2, 2, 2, 3, 3, 3, \frac{13}{4}]^{6}$	$[1,1,1,2,2,2,\frac{9}{4}]^{6}$	$[3,3,3]^{6}$	$[2,2,2]^{6}$	$[25, 14, 14, 14, 0]$	$[3, 1]$	$z^{14} + 1,z + 1$	$[1, 3, 7, 14, 32]$
2.1.16.40c1.59	$x^{16} + 2 x^{14} + 4 x^{13} + 4 x^{11} + 4 x^{9} + 8 x^{3} + 6$	$(D_4\times C_2^3):A_4$ (as 16T1037)	$768$	$2$	$[2, 2, 2, 3, 3, 3, \frac{13}{4}]^{6}$	$[1,1,1,2,2,2,\frac{9}{4}]^{6}$	$[3,3,3]^{6}$	$[2,2,2]^{6}$	$[25, 14, 14, 14, 0]$	$[3, 1]$	$z^{14} + 1,z + 1$	$[1, 3, 7, 14, 32]$
2.1.16.40c1.60	$x^{16} + 2 x^{14} + 4 x^{13} + 4 x^{11} + 4 x^{9} + 8 x^{4} + 8 x^{3} + 6$	$(D_4\times C_2^3):A_4$ (as 16T1037)	$768$	$2$	$[2, 2, 2, 3, 3, 3, \frac{13}{4}]^{6}$	$[1,1,1,2,2,2,\frac{9}{4}]^{6}$	$[3,3,3]^{6}$	$[2,2,2]^{6}$	$[25, 14, 14, 14, 0]$	$[3, 1]$	$z^{14} + 1,z + 1$	$[1, 3, 7, 14, 32]$
2.1.16.40c1.61	$x^{16} + 4 x^{15} + 2 x^{14} + 4 x^{13} + 4 x^{11} + 4 x^{9} + 6$	$C_2^4:(C_2\times A_4)$ (as 16T716)	$384$	$2$	$[2, 2, 2, 3, 3, 3, \frac{13}{4}]^{3}$	$[1,1,1,2,2,2,\frac{9}{4}]^{3}$	$[3,3,3]^{3}$	$[2,2,2]^{3}$	$[25, 14, 14, 14, 0]$	$[3, 1]$	$z^{14} + 1,z + 1$	$[1, 3, 7, 14, 32]$
2.1.16.40c1.62	$x^{16} + 4 x^{15} + 2 x^{14} + 4 x^{13} + 4 x^{11} + 4 x^{9} + 8 x^{4} + 6$	$C_2^4:(C_2\times A_4)$ (as 16T716)	$384$	$2$	$[2, 2, 2, 3, 3, 3, \frac{13}{4}]^{3}$	$[1,1,1,2,2,2,\frac{9}{4}]^{3}$	$[3,3,3]^{3}$	$[2,2,2]^{3}$	$[25, 14, 14, 14, 0]$	$[3, 1]$	$z^{14} + 1,z + 1$	$[1, 3, 7, 14, 32]$
2.1.16.40c1.63	$x^{16} + 4 x^{15} + 2 x^{14} + 4 x^{13} + 4 x^{11} + 4 x^{9} + 8 x^{3} + 6$	$C_2^4:(C_2\times A_4)$ (as 16T716)	$384$	$2$	$[2, 2, 2, 3, 3, 3, \frac{13}{4}]^{3}$	$[1,1,1,2,2,2,\frac{9}{4}]^{3}$	$[3,3,3]^{3}$	$[2,2,2]^{3}$	$[25, 14, 14, 14, 0]$	$[3, 1]$	$z^{14} + 1,z + 1$	$[1, 3, 7, 14, 32]$
2.1.16.40c1.64	$x^{16} + 4 x^{15} + 2 x^{14} + 4 x^{13} + 4 x^{11} + 4 x^{9} + 8 x^{4} + 8 x^{3} + 6$	$C_2^4:(C_2\times A_4)$ (as 16T716)	$384$	$2$	$[2, 2, 2, 3, 3, 3, \frac{13}{4}]^{3}$	$[1,1,1,2,2,2,\frac{9}{4}]^{3}$	$[3,3,3]^{3}$	$[2,2,2]^{3}$	$[25, 14, 14, 14, 0]$	$[3, 1]$	$z^{14} + 1,z + 1$	$[1, 3, 7, 14, 32]$

Download displayed columns for results to

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
Abelian varieties over $\F_{q}$
Belyi maps

Label	2.1.16.40c
Base	2.1.1.0a1.1
Degree	\(16\)
e	\(16\)
f	\(1\)
c	\(40\)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Defining polynomial

Invariants

Diagrams

Varying

Galois groups and Hidden Artin slopes

Fields

Showing all 32

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.