$p$-adic family 2.1.16.48o

• Label: 2.1.16.48o
• Base: 2.1.1.0a1.1
• Degree: \(16\)
• e: \(16\)
• f: \(1\)
• c: \(48\)

Defining polynomial

$x^{16} + 4 b_{30} x^{14} + 4 b_{28} x^{12} + 4 a_{26} x^{10} + \left(2 a_{8} + 8 c_{40}\right) x^{8} + 8 b_{39} x^{7} + 8 b_{37} x^{5} + 4 b_{20} x^{4} + 8 b_{35} x^{3} + 8 a_{33} x + 4 c_{16} + 8 c_{32} + 2$

Invariants

Residue field characteristic:	$2$
Degree:	$16$
Base field:	$\Q_{2}$
Ramification index $e$:	$16$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$48$
Artin slopes:	$[2,3,3,\frac{7}{2}]$
Swan slopes:	$[1,2,2,\frac{5}{2}]$
Means:	$\langle\frac{1}{2},\frac{5}{4},\frac{13}{8},\frac{33}{16}\rangle$
Rams:	$(1,3,3,7)$
Field count:	$152$ (complete)
Ambiguity:	$8$
Mass:	$64$
Absolute Mass:	$64$

Diagrams

Varying

Indices of inseparability:	$[33,26,20,8,0]$ (show 64), $[33,26,24,8,0]$ (show 88)
Associated inertia:	$[1,2,1]$ (show 88), $[1,3,1]$ (show 64)
Jump Set:	$[1,2,4,8,32]$ (show 76), $[1,2,13,29,45]$ (show 32), $[1,5,10,32,48]$ (show 12), $[1,5,15,31,47]$ (show 20), $[1,5,17,33,49]$ (show 12)

Galois groups and Hidden Artin slopes

Select desired size of Galois group.

		Galois groups of order 32
		$C_4^2:C_2$ (as 16T30)	$C_4:D_4$ (as 16T34)	$C_4:D_4$ (as 16T43)	$C_2^2.D_4$ (as 16T54)	Total
hidden slopes	$[\ ]^{2}$	4	4	8	8	24

		Galois groups of order 64
		$C_2^3.D_4$ (as 16T148)	$C_2\wr C_4$ (as 16T166)	Total
hidden slopes	$[\ ]^{4}$	4	4	8

		Galois groups of order 128
		$C_2^4.C_2^3$ (as 16T229)	$C_2^4.D_4$ (as 16T230)	Total
hidden slopes	$[2]^{4}$	8	16	24

		Galois groups of order 192
		$C_2\wr A_4$ (as 16T425)	$(C_2^2\times C_4):A_4$ (as 16T428)	Total
hidden slopes	$[2,2]^{3}$	8	8	16

		Galois groups of order 384
		$C_2\wr C_6$ (as 16T719)	$C_2^4:(C_2\times A_4)$ (as 16T723)	Total
hidden slopes	$[2,2,3]^{3}$	32		32
		$[2,2]^{6}$		16	16
	Total		32	16	48

		Galois groups of order 512
		$C_2^5.(C_2\times D_4)$ (as 16T833)	$C_2^5.(C_2\times D_4)$ (as 16T874)	Total
hidden slopes	$[2,2,3]^{4}$	16	16	32

Fields

Showing all 20

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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
2.1.16.48o1.13	16	$x^{16} + 4 x^{14} + 4 x^{10} + 2 x^{8} + 8 x + 2$	$C_2^5.(C_2\times D_4)$ (as 16T874)	$512$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{4}$	$[1,1,1,2,2,2,\frac{5}{2}]^{4}$	$[2,2,3]^{4}$	$[1,1,2]^{4}$	$[33, 26, 24, 8, 0]$	$[1, 2, 1]$	$z^8 + 1,z^6 + 1,z + 1$	$[1, 5, 15, 31, 47]$
2.1.16.48o1.14	16	$x^{16} + 4 x^{14} + 4 x^{10} + 10 x^{8} + 8 x + 2$	$C_2^5.(C_2\times D_4)$ (as 16T874)	$512$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{4}$	$[1,1,1,2,2,2,\frac{5}{2}]^{4}$	$[2,2,3]^{4}$	$[1,1,2]^{4}$	$[33, 26, 24, 8, 0]$	$[1, 2, 1]$	$z^8 + 1,z^6 + 1,z + 1$	$[1, 5, 15, 31, 47]$
2.1.16.48o1.15	16	$x^{16} + 4 x^{14} + 4 x^{10} + 2 x^{8} + 8 x^{5} + 8 x + 2$	$C_2^5.(C_2\times D_4)$ (as 16T874)	$512$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{4}$	$[1,1,1,2,2,2,\frac{5}{2}]^{4}$	$[2,2,3]^{4}$	$[1,1,2]^{4}$	$[33, 26, 24, 8, 0]$	$[1, 2, 1]$	$z^8 + 1,z^6 + 1,z + 1$	$[1, 5, 15, 31, 47]$
2.1.16.48o1.16	16	$x^{16} + 4 x^{14} + 4 x^{10} + 10 x^{8} + 8 x^{5} + 8 x + 2$	$C_2^5.(C_2\times D_4)$ (as 16T874)	$512$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{4}$	$[1,1,1,2,2,2,\frac{5}{2}]^{4}$	$[2,2,3]^{4}$	$[1,1,2]^{4}$	$[33, 26, 24, 8, 0]$	$[1, 2, 1]$	$z^8 + 1,z^6 + 1,z + 1$	$[1, 5, 15, 31, 47]$
2.1.16.48o1.17	16	$x^{16} + 4 x^{14} + 4 x^{10} + 2 x^{8} + 8 x^{3} + 8 x + 2$	$C_2^4.D_4$ (as 16T230)	$128$	$4$	$[2, 2, 3, 3, \frac{7}{2}]^{4}$	$[1,1,2,2,\frac{5}{2}]^{4}$	$[2]^{4}$	$[1]^{4}$	$[33, 26, 24, 8, 0]$	$[1, 2, 1]$	$z^8 + 1,z^6 + 1,z + 1$	$[1, 5, 15, 31, 47]$
2.1.16.48o1.18	16	$x^{16} + 4 x^{14} + 4 x^{10} + 10 x^{8} + 8 x^{3} + 8 x + 2$	$C_2^4.D_4$ (as 16T230)	$128$	$4$	$[2, 2, 3, 3, \frac{7}{2}]^{4}$	$[1,1,2,2,\frac{5}{2}]^{4}$	$[2]^{4}$	$[1]^{4}$	$[33, 26, 24, 8, 0]$	$[1, 2, 1]$	$z^8 + 1,z^6 + 1,z + 1$	$[1, 5, 15, 31, 47]$
2.1.16.48o1.19	16	$x^{16} + 4 x^{14} + 4 x^{10} + 2 x^{8} + 8 x^{7} + 8 x^{3} + 8 x + 2$	$C_2^4.D_4$ (as 16T230)	$128$	$4$	$[2, 2, 3, 3, \frac{7}{2}]^{4}$	$[1,1,2,2,\frac{5}{2}]^{4}$	$[2]^{4}$	$[1]^{4}$	$[33, 26, 24, 8, 0]$	$[1, 2, 1]$	$z^8 + 1,z^6 + 1,z + 1$	$[1, 5, 15, 31, 47]$
2.1.16.48o1.20	16	$x^{16} + 4 x^{14} + 4 x^{10} + 10 x^{8} + 8 x^{7} + 8 x^{3} + 8 x + 2$	$C_2^4.D_4$ (as 16T230)	$128$	$4$	$[2, 2, 3, 3, \frac{7}{2}]^{4}$	$[1,1,2,2,\frac{5}{2}]^{4}$	$[2]^{4}$	$[1]^{4}$	$[33, 26, 24, 8, 0]$	$[1, 2, 1]$	$z^8 + 1,z^6 + 1,z + 1$	$[1, 5, 15, 31, 47]$
2.1.16.48o1.21	4	$x^{16} + 4 x^{14} + 4 x^{10} + 2 x^{8} + 8 x^{5} + 8 x^{3} + 8 x + 2$	$C_2^3.D_4$ (as 16T148)	$64$	$2$	$[2, 3, 3, \frac{7}{2}]^{4}$	$[1,2,2,\frac{5}{2}]^{4}$	$[\ ]^{4}$	$[\ ]^{4}$	$[33, 26, 24, 8, 0]$	$[1, 2, 1]$	$z^8 + 1,z^6 + 1,z + 1$	$[1, 5, 15, 31, 47]$
2.1.16.48o1.22	4	$x^{16} + 4 x^{14} + 4 x^{10} + 2 x^{8} + 8 x^{7} + 8 x^{5} + 8 x^{3} + 8 x + 2$	$C_2^3.D_4$ (as 16T148)	$64$	$2$	$[2, 3, 3, \frac{7}{2}]^{4}$	$[1,2,2,\frac{5}{2}]^{4}$	$[\ ]^{4}$	$[\ ]^{4}$	$[33, 26, 24, 8, 0]$	$[1, 2, 1]$	$z^8 + 1,z^6 + 1,z + 1$	$[1, 5, 15, 31, 47]$
2.1.16.48o1.23	4	$x^{16} + 4 x^{12} + 4 x^{10} + 2 x^{8} + 8 x + 2$	$C_2\wr C_4$ (as 16T166)	$64$	$2$	$[2, 3, 3, \frac{7}{2}]^{4}$	$[1,2,2,\frac{5}{2}]^{4}$	$[\ ]^{4}$	$[\ ]^{4}$	$[33, 26, 24, 8, 0]$	$[1, 2, 1]$	$z^8 + 1,z^6 + 1,z + 1$	$[1, 5, 15, 31, 47]$
2.1.16.48o1.24	4	$x^{16} + 4 x^{12} + 4 x^{10} + 2 x^{8} + 8 x^{7} + 8 x + 2$	$C_2\wr C_4$ (as 16T166)	$64$	$2$	$[2, 3, 3, \frac{7}{2}]^{4}$	$[1,2,2,\frac{5}{2}]^{4}$	$[\ ]^{4}$	$[\ ]^{4}$	$[33, 26, 24, 8, 0]$	$[1, 2, 1]$	$z^8 + 1,z^6 + 1,z + 1$	$[1, 5, 15, 31, 47]$
2.1.16.48o1.25	16	$x^{16} + 4 x^{12} + 4 x^{10} + 2 x^{8} + 8 x^{5} + 8 x + 2$	$C_2^4.D_4$ (as 16T230)	$128$	$4$	$[2, 2, 3, 3, \frac{7}{2}]^{4}$	$[1,1,2,2,\frac{5}{2}]^{4}$	$[2]^{4}$	$[1]^{4}$	$[33, 26, 24, 8, 0]$	$[1, 2, 1]$	$z^8 + 1,z^6 + 1,z + 1$	$[1, 5, 15, 31, 47]$
2.1.16.48o1.26	16	$x^{16} + 4 x^{12} + 4 x^{10} + 10 x^{8} + 8 x^{5} + 8 x + 2$	$C_2^4.D_4$ (as 16T230)	$128$	$4$	$[2, 2, 3, 3, \frac{7}{2}]^{4}$	$[1,1,2,2,\frac{5}{2}]^{4}$	$[2]^{4}$	$[1]^{4}$	$[33, 26, 24, 8, 0]$	$[1, 2, 1]$	$z^8 + 1,z^6 + 1,z + 1$	$[1, 5, 15, 31, 47]$
2.1.16.48o1.27	16	$x^{16} + 4 x^{12} + 4 x^{10} + 2 x^{8} + 8 x^{7} + 8 x^{5} + 8 x + 2$	$C_2^4.D_4$ (as 16T230)	$128$	$4$	$[2, 2, 3, 3, \frac{7}{2}]^{4}$	$[1,1,2,2,\frac{5}{2}]^{4}$	$[2]^{4}$	$[1]^{4}$	$[33, 26, 24, 8, 0]$	$[1, 2, 1]$	$z^8 + 1,z^6 + 1,z + 1$	$[1, 5, 15, 31, 47]$
2.1.16.48o1.28	16	$x^{16} + 4 x^{12} + 4 x^{10} + 10 x^{8} + 8 x^{7} + 8 x^{5} + 8 x + 2$	$C_2^4.D_4$ (as 16T230)	$128$	$4$	$[2, 2, 3, 3, \frac{7}{2}]^{4}$	$[1,1,2,2,\frac{5}{2}]^{4}$	$[2]^{4}$	$[1]^{4}$	$[33, 26, 24, 8, 0]$	$[1, 2, 1]$	$z^8 + 1,z^6 + 1,z + 1$	$[1, 5, 15, 31, 47]$
2.1.16.48o1.29	16	$x^{16} + 4 x^{12} + 4 x^{10} + 2 x^{8} + 8 x^{3} + 8 x + 2$	$C_2^5.(C_2\times D_4)$ (as 16T874)	$512$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{4}$	$[1,1,1,2,2,2,\frac{5}{2}]^{4}$	$[2,2,3]^{4}$	$[1,1,2]^{4}$	$[33, 26, 24, 8, 0]$	$[1, 2, 1]$	$z^8 + 1,z^6 + 1,z + 1$	$[1, 5, 15, 31, 47]$
2.1.16.48o1.30	16	$x^{16} + 4 x^{12} + 4 x^{10} + 10 x^{8} + 8 x^{3} + 8 x + 2$	$C_2^5.(C_2\times D_4)$ (as 16T874)	$512$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{4}$	$[1,1,1,2,2,2,\frac{5}{2}]^{4}$	$[2,2,3]^{4}$	$[1,1,2]^{4}$	$[33, 26, 24, 8, 0]$	$[1, 2, 1]$	$z^8 + 1,z^6 + 1,z + 1$	$[1, 5, 15, 31, 47]$
2.1.16.48o1.31	16	$x^{16} + 4 x^{12} + 4 x^{10} + 2 x^{8} + 8 x^{5} + 8 x^{3} + 8 x + 2$	$C_2^5.(C_2\times D_4)$ (as 16T874)	$512$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{4}$	$[1,1,1,2,2,2,\frac{5}{2}]^{4}$	$[2,2,3]^{4}$	$[1,1,2]^{4}$	$[33, 26, 24, 8, 0]$	$[1, 2, 1]$	$z^8 + 1,z^6 + 1,z + 1$	$[1, 5, 15, 31, 47]$
2.1.16.48o1.32	16	$x^{16} + 4 x^{12} + 4 x^{10} + 10 x^{8} + 8 x^{5} + 8 x^{3} + 8 x + 2$	$C_2^5.(C_2\times D_4)$ (as 16T874)	$512$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{4}$	$[1,1,1,2,2,2,\frac{5}{2}]^{4}$	$[2,2,3]^{4}$	$[1,1,2]^{4}$	$[33, 26, 24, 8, 0]$	$[1, 2, 1]$	$z^8 + 1,z^6 + 1,z + 1$	$[1, 5, 15, 31, 47]$

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