$p$-adic family 2.1.16.54o

• Label: 2.1.16.54o
• Base: 2.1.1.0a1.1
• Degree: \(16\)
• e: \(16\)
• f: \(1\)
• c: \(54\)

Defining polynomial

$x^{16} + 8 b_{47} x^{15} + 4 a_{30} x^{14} + 8 b_{45} x^{13} + 4 b_{28} x^{12} + 8 b_{43} x^{11} + 8 b_{41} x^{9} + \left(2 a_{8} + 8 c_{40}\right) x^{8} + 8 a_{39} x^{7} + 8 b_{38} x^{6} + 4 a_{20} x^{4} + 8 b_{34} x^{2} + 4 c_{16} + 8 c_{32} + 16 c_{48} + 2$

Invariants

Residue field characteristic:	$2$
Degree:	$16$
Base field:	$\Q_{2}$
Ramification index $e$:	$16$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$54$
Artin slopes:	$[2,3,\frac{7}{2},4]$
Swan slopes:	$[1,2,\frac{5}{2},3]$
Means:	$\langle\frac{1}{2},\frac{5}{4},\frac{15}{8},\frac{39}{16}\rangle$
Rams:	$(1,3,5,9)$
Field count:	$432$ (complete)
Ambiguity:	$16$
Mass:	$128$
Absolute Mass:	$128$

Diagrams

Varying

Indices of inseparability:	$[39,30,20,8,0]$
Associated inertia:	$[1,1,1,1]$
Jump Set:	$[1,2,4,8,32]$ (show 216), $[1,2,15,31,47]$ (show 96), $[1,7,14,28,44]$ (show 5), $[1,7,14,32,48]$ (show 60), $[1,7,25,41,57]$ (show 20), $[1,7,27,43,59]$ (show 12), $[1,7,29,45,61]$ (show 12), $[1,7,31,47,63]$ (show 6), $[1,7,32,48,64]$ (show 5)

Galois groups and Hidden Artin slopes

Select desired size of Galois group.

		Galois groups of order 16
		$C_4:C_4$ (as 16T8)	$C_2^2:C_4$ (as 16T10)	Total
hidden slopes	$[\ ]$	8	8	16

		Galois groups of order 32
		$C_4^2:C_2$ (as 16T17)	$C_4\times D_4$ (as 16T19)	$C_2^3:C_4$ (as 16T21)	$C_4^2:C_2$ (as 16T27)	$C_4^2:C_2$ (as 16T30)	$C_2^2:Q_8$ (as 16T31)	$C_4:D_4$ (as 16T34)	$C_2^2.D_4$ (as 16T37)	$C_2^2\wr C_2$ (as 16T39)	$C_4:D_4$ (as 16T43)	$C_2^2\wr C_2$ (as 16T46)	$C_2^2.D_4$ (as 16T54)	Total
hidden slopes	$[\ ]^{2}$	4	16	4	4	4	12	8	6	8	12	2	16	96

		Galois groups of order 64
		$D_4:D_4$ (as 16T126)	$C_2\wr C_2^2$ (as 16T129)	$C_2^3.D_4$ (as 16T146)	$C_2\wr C_2^2$ (as 16T147)	$C_2^3.D_4$ (as 16T148)	$C_2\wr C_2^2$ (as 16T150)	$C_2^2:\SD_{16}$ (as 16T155)	$C_2^2.Q_{16}$ (as 16T161)	$C_2^2.D_8$ (as 16T163)	$C_2^3.D_4$ (as 16T165)	$C_2\wr C_4$ (as 16T166)	$C_2^3.D_4$ (as 16T177)	Total
hidden slopes	$[2]^{2}$	8	8	4	4	4	8	8	8	8	8	4	8	80

		Galois groups of order 128
		$C_4^2:D_4$ (as 16T211)	$C_2^4.D_4$ (as 16T218)	$C_2^4.D_4$ (as 16T254)	$C_2^4.D_4$ (as 16T318)	$C_2^4.D_4$ (as 16T319)	$C_2^4.D_4$ (as 16T326)	$C_4^2:D_4$ (as 16T340)	$C_4^2.D_4$ (as 16T344)	$C_4^2:D_4$ (as 16T352)	$C_4^2:D_4$ (as 16T366)	$C_4^2.D_4$ (as 16T380)	$C_4^2:D_4$ (as 16T400)	$C_4^2:D_4$ (as 16T402)	$C_4^2:D_4$ (as 16T410)	Total
hidden slopes	$[2,\frac{7}{2}]^{2}$							8	16	8	8	16	16	8	16	96
		$[2]^{4}$	32	8	16	8	8	8									80
	Total		32	8	16	8	8	8	8	16	8	8	16	16	8	16	176

		Galois groups of order 512
		$C_2^4.(C_4\times D_4)$ (as 16T824)	$C_2^4.(C_4\times D_4)$ (as 16T896)	$C_2^4.(C_4\times D_4)$ (as 16T926)	$C_2^4.(C_4\times D_4)$ (as 16T930)	Total
hidden slopes	$[2,2,\frac{7}{2}]^{4}$	16	16	16	16	64

Fields

Showing all 6

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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
2.1.16.54o1.103	4	$x^{16} + 4 x^{14} + 8 x^{13} + 4 x^{12} + 8 x^{9} + 2 x^{8} + 8 x^{7} + 4 x^{4} + 2$	$C_4^2:C_2$ (as 16T27)	$32$	$4$	$[2, 3, \frac{7}{2}, 4]^{2}$	$[1,2,\frac{5}{2},3]^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[39, 30, 20, 8, 0]$	$[1, 1, 1, 1]$	$z^8 + 1,z^4 + 1,z^2 + 1,z + 1$	$[1, 7, 31, 47, 63]$
2.1.16.54o1.119	6	$x^{16} + 4 x^{14} + 8 x^{13} + 4 x^{12} + 8 x^{11} + 8 x^{9} + 2 x^{8} + 8 x^{7} + 8 x^{6} + 4 x^{4} + 2$	$C_2^2.D_4$ (as 16T37)	$32$	$4$	$[2, 3, \frac{7}{2}, 4]^{2}$	$[1,2,\frac{5}{2},3]^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[39, 30, 20, 8, 0]$	$[1, 1, 1, 1]$	$z^8 + 1,z^4 + 1,z^2 + 1,z + 1$	$[1, 7, 31, 47, 63]$
2.1.16.54o1.125	12	$x^{16} + 4 x^{14} + 8 x^{13} + 4 x^{12} + 2 x^{8} + 8 x^{7} + 4 x^{4} + 8 x^{2} + 2$	$C_4:D_4$ (as 16T43)	$32$	$8$	$[2, 3, \frac{7}{2}, 4]^{2}$	$[1,2,\frac{5}{2},3]^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[39, 30, 20, 8, 0]$	$[1, 1, 1, 1]$	$z^8 + 1,z^4 + 1,z^2 + 1,z + 1$	$[1, 7, 31, 47, 63]$
2.1.16.54o1.135	12	$x^{16} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 4 x^{12} + 10 x^{8} + 8 x^{7} + 4 x^{4} + 8 x^{2} + 2$	$C_4:D_4$ (as 16T43)	$32$	$8$	$[2, 3, \frac{7}{2}, 4]^{2}$	$[1,2,\frac{5}{2},3]^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[39, 30, 20, 8, 0]$	$[1, 1, 1, 1]$	$z^8 + 1,z^4 + 1,z^2 + 1,z + 1$	$[1, 7, 31, 47, 63]$
2.1.16.54o1.141	16	$x^{16} + 4 x^{14} + 8 x^{13} + 4 x^{12} + 8 x^{11} + 2 x^{8} + 8 x^{7} + 8 x^{6} + 4 x^{4} + 8 x^{2} + 2$	$C_2^2.D_4$ (as 16T54)	$32$	$8$	$[2, 3, \frac{7}{2}, 4]^{2}$	$[1,2,\frac{5}{2},3]^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[39, 30, 20, 8, 0]$	$[1, 1, 1, 1]$	$z^8 + 1,z^4 + 1,z^2 + 1,z + 1$	$[1, 7, 31, 47, 63]$
2.1.16.54o1.153	16	$x^{16} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 4 x^{12} + 8 x^{11} + 10 x^{8} + 8 x^{7} + 8 x^{6} + 4 x^{4} + 8 x^{2} + 2$	$C_2^2.D_4$ (as 16T54)	$32$	$8$	$[2, 3, \frac{7}{2}, 4]^{2}$	$[1,2,\frac{5}{2},3]^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[39, 30, 20, 8, 0]$	$[1, 1, 1, 1]$	$z^8 + 1,z^4 + 1,z^2 + 1,z + 1$	$[1, 7, 31, 47, 63]$

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