$p$-adic family 2.1.16.56l

• Label: 2.1.16.56l
• Base: 2.1.1.0a1.1
• Degree: \(16\)
• e: \(16\)
• f: \(1\)
• c: \(56\)

Defining polynomial

$x^{16} + 8 b_{47} x^{15} + 8 b_{46} x^{14} + 8 b_{45} x^{13} + 4 b_{28} x^{12} + 8 b_{43} x^{11} + 8 b_{42} x^{10} + 8 a_{41} x^{9} + 2 a_{8} x^{8} + 8 b_{38} x^{6} + 4 a_{20} x^{4} + 8 b_{34} x^{2} + 4 c_{16} + 8 c_{32} + 16 c_{48} + 2$

Invariants

Residue field characteristic:	$2$
Degree:	$16$
Base field:	$\Q_{2}$
Ramification index $e$:	$16$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$56$
Artin slopes:	$[2,3,4,4]$
Swan slopes:	$[1,2,3,3]$
Means:	$\langle\frac{1}{2},\frac{5}{4},\frac{17}{8},\frac{41}{16}\rangle$
Rams:	$(1,3,7,7)$
Field count:	$384$ (complete)
Ambiguity:	$8$
Mass:	$256$
Absolute Mass:	$256$

Diagrams

Varying

Indices of inseparability:	$[41,34,20,8,0]$ (show 128), $[41,36,20,8,0]$ (show 256)
Associated inertia:	$[1,1,2]$ (show 256), $[1,1,3]$ (show 128)
Jump Set:	$[1,2,4,8,32]$ (show 192), $[1,2,4,32,48]$ (show 48), $[1,2,25,41,57]$ (show 8), $[1,7,15,31,47]$ (show 96), $[1,9,18,34,50]$ (show 2), $[1,9,27,43,59]$ (show 16), $[1,9,29,45,61]$ (show 8), $[1,9,31,47,63]$ (show 4), $[1,9,32,48,64]$ (show 2), $[1,11,25,41,57]$ (show 8)

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 128
		$C_2^4.C_2^3$ (as 16T229)	$C_2^4.C_2^3$ (as 16T231)	$C_2^4.D_4$ (as 16T268)	$C_2^4.D_4$ (as 16T284)	$C_2^4.C_2^3$ (as 16T316)	$C_2^4.D_4$ (as 16T318)	$C_2^4.D_4$ (as 16T326)	$C_2^4:Q_8$ (as 16T354)	$C_2^4:Q_8$ (as 16T368)	$C_2^4.C_2^3$ (as 16T383)	$C_2^4:Q_8$ (as 16T384)	Total
hidden slopes	$[3,\frac{7}{2}]^{2}$	16	8	8	8	8	8	8	8	16	32	8	128

		Galois groups of order 256
		$C_2^4.D_8$ (as 16T686)	$C_2^4.Q_{16}$ (as 16T691)	$C_2^4.Q_{16}$ (as 16T698)	$C_2^4.D_8$ (as 16T705)	Total
hidden slopes	$[3,\frac{7}{2}]^{4}$	16	16	16	16	64

		Galois groups of order 512
		$C_2^6.D_4$ (as 16T900)	$C_2^6.D_4$ (as 16T921)	Total
hidden slopes	$[2,3,\frac{7}{2}]^{4}$	32	32	64

		Galois groups of order 3072
		$C_2^8:(C_2\times C_6)$ (as 16T1516)
hidden slopes	$[2,2,3,3,\frac{7}{2},\frac{7}{2}]^{3}$	64

		Galois groups of order 6144
		$C_2^6.(D_4\times A_4)$ (as 16T1655)
hidden slopes	$[2,2,3,3,\frac{7}{2},\frac{7}{2}]^{6}$	8

Fields

Showing all 8

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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
2.1.16.56l1.7	16	$x^{16} + 8 x^{13} + 8 x^{9} + 2 x^{8} + 4 x^{4} + 2$	$C_2^4.C_2^3$ (as 16T229)	$128$	$2$	$[2, 3, 3, \frac{7}{2}, 4, 4]^{2}$	$[1,2,2,\frac{5}{2},3,3]^{2}$	$[3,\frac{7}{2}]^{2}$	$[2,\frac{5}{2}]^{2}$	$[41, 36, 20, 8, 0]$	$[1, 1, 2]$	$z^8 + 1,z^4 + 1,z^3 + 1$	$[1, 9, 29, 45, 61]$
2.1.16.56l1.8	16	$x^{16} + 8 x^{15} + 8 x^{13} + 8 x^{9} + 2 x^{8} + 4 x^{4} + 2$	$C_2^4:Q_8$ (as 16T368)	$128$	$2$	$[2, 3, 3, \frac{7}{2}, 4, 4]^{2}$	$[1,2,2,\frac{5}{2},3,3]^{2}$	$[3,\frac{7}{2}]^{2}$	$[2,\frac{5}{2}]^{2}$	$[41, 36, 20, 8, 0]$	$[1, 1, 2]$	$z^8 + 1,z^4 + 1,z^3 + 1$	$[1, 9, 29, 45, 61]$
2.1.16.56l1.9	16	$x^{16} + 8 x^{14} + 8 x^{13} + 8 x^{9} + 2 x^{8} + 4 x^{4} + 2$	$C_2^4:Q_8$ (as 16T368)	$128$	$2$	$[2, 3, 3, \frac{7}{2}, 4, 4]^{2}$	$[1,2,2,\frac{5}{2},3,3]^{2}$	$[3,\frac{7}{2}]^{2}$	$[2,\frac{5}{2}]^{2}$	$[41, 36, 20, 8, 0]$	$[1, 1, 2]$	$z^8 + 1,z^4 + 1,z^3 + 1$	$[1, 9, 29, 45, 61]$
2.1.16.56l1.10	16	$x^{16} + 8 x^{15} + 8 x^{14} + 8 x^{13} + 8 x^{9} + 2 x^{8} + 4 x^{4} + 2$	$C_2^4.C_2^3$ (as 16T229)	$128$	$2$	$[2, 3, 3, \frac{7}{2}, 4, 4]^{2}$	$[1,2,2,\frac{5}{2},3,3]^{2}$	$[3,\frac{7}{2}]^{2}$	$[2,\frac{5}{2}]^{2}$	$[41, 36, 20, 8, 0]$	$[1, 1, 2]$	$z^8 + 1,z^4 + 1,z^3 + 1$	$[1, 9, 29, 45, 61]$
2.1.16.56l1.21	16	$x^{16} + 8 x^{10} + 8 x^{9} + 2 x^{8} + 4 x^{4} + 2$	$C_2^4.C_2^3$ (as 16T229)	$128$	$2$	$[2, 3, 3, \frac{7}{2}, 4, 4]^{2}$	$[1,2,2,\frac{5}{2},3,3]^{2}$	$[3,\frac{7}{2}]^{2}$	$[2,\frac{5}{2}]^{2}$	$[41, 36, 20, 8, 0]$	$[1, 1, 2]$	$z^8 + 1,z^4 + 1,z^3 + 1$	$[1, 9, 29, 45, 61]$
2.1.16.56l1.22	16	$x^{16} + 8 x^{15} + 8 x^{10} + 8 x^{9} + 2 x^{8} + 4 x^{4} + 2$	$C_2^4:Q_8$ (as 16T368)	$128$	$2$	$[2, 3, 3, \frac{7}{2}, 4, 4]^{2}$	$[1,2,2,\frac{5}{2},3,3]^{2}$	$[3,\frac{7}{2}]^{2}$	$[2,\frac{5}{2}]^{2}$	$[41, 36, 20, 8, 0]$	$[1, 1, 2]$	$z^8 + 1,z^4 + 1,z^3 + 1$	$[1, 9, 29, 45, 61]$
2.1.16.56l1.23	16	$x^{16} + 8 x^{14} + 8 x^{10} + 8 x^{9} + 2 x^{8} + 4 x^{4} + 2$	$C_2^4:Q_8$ (as 16T368)	$128$	$2$	$[2, 3, 3, \frac{7}{2}, 4, 4]^{2}$	$[1,2,2,\frac{5}{2},3,3]^{2}$	$[3,\frac{7}{2}]^{2}$	$[2,\frac{5}{2}]^{2}$	$[41, 36, 20, 8, 0]$	$[1, 1, 2]$	$z^8 + 1,z^4 + 1,z^3 + 1$	$[1, 9, 29, 45, 61]$
2.1.16.56l1.24	16	$x^{16} + 8 x^{15} + 8 x^{14} + 8 x^{10} + 8 x^{9} + 2 x^{8} + 4 x^{4} + 2$	$C_2^4.C_2^3$ (as 16T229)	$128$	$2$	$[2, 3, 3, \frac{7}{2}, 4, 4]^{2}$	$[1,2,2,\frac{5}{2},3,3]^{2}$	$[3,\frac{7}{2}]^{2}$	$[2,\frac{5}{2}]^{2}$	$[41, 36, 20, 8, 0]$	$[1, 1, 2]$	$z^8 + 1,z^4 + 1,z^3 + 1$	$[1, 9, 29, 45, 61]$

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