$p$-adic family 3.1.18.37d

• Label: 3.1.18.37d
• Base: 3.1.1.0a1.1
• Degree: \(18\)
• e: \(18\)
• f: \(1\)
• c: \(37\)

Defining polynomial

$x^{18} + 3 b_{15} x^{15} + 3 a_{12} x^{12} + 9 c_{24} x^{6} + 9 b_{23} x^{5} + 9 b_{22} x^{4} + 9 a_{20} x^{2} + 3 d_{0} + 9 c_{18}$

Invariants

Residue field characteristic:	$3$
Degree:	$18$
Base field:	$\Q_{3}$
Ramification index $e$:	$18$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$37$
Artin slopes:	$[2,\frac{7}{3}]$
Swan slopes:	$[1,\frac{4}{3}]$
Means:	$\langle\frac{2}{3},\frac{10}{9}\rangle$
Rams:	$(2,4)$
Field count:	$240$ (complete)
Ambiguity:	$18$
Mass:	$108$
Absolute Mass:	$108$

Diagrams

Varying

Indices of inseparability:	$[20,12,0]$
Associated inertia:	$[1,1,1]$ (show 90), $[1,1,2]$ (show 30), $[1,2,1]$ (show 90), $[1,2,2]$ (show 30)
Jump Set:	undefined (show 120), $[1,3,27]$ (show 16), $[1,7,25]$ (show 60), $[1,8,26]$ (show 36), $[1,11,29]$ (show 8)

Fields

Showing 1-50 of 120

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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
3.1.18.37d1.16		$x^{18} + 3 x^{12} + 18 x^{2} + 6$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^6 + 1,z^2 + 1$	undefined
3.1.18.37d1.17		$x^{18} + 3 x^{12} + 9 x^{5} + 18 x^{2} + 6$	$C_3^2:S_3^2$ (as 18T124)	$324$	$1$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^6 + 1,z^2 + 1$	undefined
3.1.18.37d1.18		$x^{18} + 3 x^{12} + 18 x^{2} + 15$	$C_3^2:D_6$ (as 18T42)	$108$	$2$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^6 + 1,z^2 + 1$	undefined
3.1.18.37d1.19		$x^{18} + 3 x^{12} + 9 x^{5} + 18 x^{2} + 15$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^6 + 1,z^2 + 1$	undefined
3.1.18.37d1.20		$x^{18} + 3 x^{12} + 18 x^{2} + 24$	not computed	$ $not computed$ $	$2$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^6 + 1,z^2 + 1$	undefined
3.1.18.37d1.21		$x^{18} + 3 x^{12} + 9 x^{5} + 18 x^{2} + 24$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^6 + 1,z^2 + 1$	undefined
3.1.18.37d1.22		$x^{18} + 3 x^{15} + 3 x^{12} + 18 x^{2} + 6$	$C_3^4:S_3^2$ (as 18T409)	$2916$	$1$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^6 + 1,z^2 + 1$	undefined
3.1.18.37d1.23		$x^{18} + 3 x^{15} + 3 x^{12} + 9 x^{5} + 18 x^{2} + 6$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^6 + 1,z^2 + 1$	undefined
3.1.18.37d1.24		$x^{18} + 3 x^{15} + 3 x^{12} + 18 x^{2} + 15$	$C_3^4:S_3^2$ (as 18T409)	$2916$	$1$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^6 + 1,z^2 + 1$	undefined
3.1.18.37d1.25		$x^{18} + 3 x^{15} + 3 x^{12} + 9 x^{5} + 18 x^{2} + 15$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^6 + 1,z^2 + 1$	undefined
3.1.18.37d1.26		$x^{18} + 3 x^{15} + 3 x^{12} + 18 x^{2} + 24$	$C_3^4:S_3^2$ (as 18T409)	$2916$	$1$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^6 + 1,z^2 + 1$	undefined
3.1.18.37d1.27		$x^{18} + 3 x^{15} + 3 x^{12} + 9 x^{5} + 18 x^{2} + 24$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^6 + 1,z^2 + 1$	undefined
3.1.18.37d1.28		$x^{18} + 6 x^{15} + 3 x^{12} + 9 x^{5} + 18 x^{2} + 6$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^6 + 1,z^2 + 1$	undefined
3.1.18.37d1.29		$x^{18} + 6 x^{15} + 3 x^{12} + 9 x^{5} + 18 x^{2} + 15$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^6 + 1,z^2 + 1$	undefined
3.1.18.37d1.30		$x^{18} + 6 x^{15} + 3 x^{12} + 9 x^{5} + 18 x^{2} + 24$	$C_3^4:S_3^2$ (as 18T409)	$2916$	$1$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^6 + 1,z^2 + 1$	undefined
3.1.18.37d2.46		$x^{18} + 3 x^{12} + 9 x^{2} + 6$	not computed	$ $not computed$ $	$6$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^6 + 1,z^2 + 2$	undefined
3.1.18.37d2.47		$x^{18} + 3 x^{12} + 9 x^{6} + 9 x^{2} + 6$	$C_3^3:C_6$ (as 18T75)	$162$	$6$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^6 + 1,z^2 + 2$	undefined
3.1.18.37d2.48		$x^{18} + 3 x^{12} + 18 x^{6} + 9 x^{2} + 6$	$C_3^3:C_6$ (as 18T75)	$162$	$6$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^6 + 1,z^2 + 2$	undefined
3.1.18.37d2.49		$x^{18} + 3 x^{12} + 9 x^{5} + 9 x^{2} + 6$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^6 + 1,z^2 + 2$	undefined
3.1.18.37d2.50		$x^{18} + 3 x^{12} + 9 x^{6} + 9 x^{5} + 9 x^{2} + 6$	$C_3^4:C_6$ (as 18T163)	$486$	$3$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^6 + 1,z^2 + 2$	undefined
3.1.18.37d2.51		$x^{18} + 3 x^{12} + 18 x^{6} + 9 x^{5} + 9 x^{2} + 6$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^6 + 1,z^2 + 2$	undefined
3.1.18.37d2.52		$x^{18} + 3 x^{12} + 9 x^{2} + 15$	$C_3^3:C_6$ (as 18T75)	$162$	$6$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^6 + 1,z^2 + 2$	undefined
3.1.18.37d2.53		$x^{18} + 3 x^{12} + 9 x^{6} + 9 x^{2} + 15$	$C_3^3:C_6$ (as 18T75)	$162$	$6$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^6 + 1,z^2 + 2$	undefined
3.1.18.37d2.54		$x^{18} + 3 x^{12} + 18 x^{6} + 9 x^{2} + 15$	not computed	$ $not computed$ $	$6$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^6 + 1,z^2 + 2$	undefined
3.1.18.37d2.55		$x^{18} + 3 x^{12} + 9 x^{5} + 9 x^{2} + 15$	$C_3^4:C_6$ (as 18T163)	$486$	$3$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^6 + 1,z^2 + 2$	undefined
3.1.18.37d2.56		$x^{18} + 3 x^{12} + 9 x^{6} + 9 x^{5} + 9 x^{2} + 15$	$C_3^4:C_6$ (as 18T163)	$486$	$3$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^6 + 1,z^2 + 2$	undefined
3.1.18.37d2.57		$x^{18} + 3 x^{12} + 18 x^{6} + 9 x^{5} + 9 x^{2} + 15$	$C_3^4:C_6$ (as 18T163)	$486$	$3$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^6 + 1,z^2 + 2$	undefined
3.1.18.37d2.58		$x^{18} + 3 x^{12} + 9 x^{2} + 24$	$C_3^3:C_6$ (as 18T75)	$162$	$6$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^6 + 1,z^2 + 2$	undefined
3.1.18.37d2.59		$x^{18} + 3 x^{12} + 9 x^{6} + 9 x^{2} + 24$	not computed	$ $not computed$ $	$6$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^6 + 1,z^2 + 2$	undefined
3.1.18.37d2.60		$x^{18} + 3 x^{12} + 18 x^{6} + 9 x^{2} + 24$	not computed	$ $not computed$ $	$6$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^6 + 1,z^2 + 2$	undefined
3.1.18.37d2.61		$x^{18} + 3 x^{12} + 9 x^{5} + 9 x^{2} + 24$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^6 + 1,z^2 + 2$	undefined
3.1.18.37d2.62		$x^{18} + 3 x^{12} + 9 x^{6} + 9 x^{5} + 9 x^{2} + 24$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^6 + 1,z^2 + 2$	undefined
3.1.18.37d2.63		$x^{18} + 3 x^{12} + 18 x^{6} + 9 x^{5} + 9 x^{2} + 24$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^6 + 1,z^2 + 2$	undefined
3.1.18.37d2.64		$x^{18} + 3 x^{15} + 3 x^{12} + 9 x^{2} + 6$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^6 + 1,z^2 + 2$	undefined
3.1.18.37d2.65		$x^{18} + 3 x^{15} + 3 x^{12} + 9 x^{6} + 9 x^{2} + 6$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^6 + 1,z^2 + 2$	undefined
3.1.18.37d2.66		$x^{18} + 3 x^{15} + 3 x^{12} + 18 x^{6} + 9 x^{2} + 6$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^6 + 1,z^2 + 2$	undefined
3.1.18.37d2.67		$x^{18} + 3 x^{15} + 3 x^{12} + 9 x^{5} + 9 x^{2} + 6$	$\He_3^2:C_6$ (as 18T455)	$4374$	$3$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^6 + 1,z^2 + 2$	undefined
3.1.18.37d2.68		$x^{18} + 3 x^{15} + 3 x^{12} + 9 x^{6} + 9 x^{5} + 9 x^{2} + 6$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^6 + 1,z^2 + 2$	undefined
3.1.18.37d2.69		$x^{18} + 3 x^{15} + 3 x^{12} + 18 x^{6} + 9 x^{5} + 9 x^{2} + 6$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^6 + 1,z^2 + 2$	undefined
3.1.18.37d2.70		$x^{18} + 3 x^{15} + 3 x^{12} + 9 x^{2} + 15$	$\He_3^2:C_6$ (as 18T455)	$4374$	$3$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^6 + 1,z^2 + 2$	undefined
3.1.18.37d2.71		$x^{18} + 3 x^{15} + 3 x^{12} + 9 x^{6} + 9 x^{2} + 15$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^6 + 1,z^2 + 2$	undefined
3.1.18.37d2.72		$x^{18} + 3 x^{15} + 3 x^{12} + 18 x^{6} + 9 x^{2} + 15$	$\He_3^2:C_6$ (as 18T455)	$4374$	$3$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^6 + 1,z^2 + 2$	undefined
3.1.18.37d2.73		$x^{18} + 3 x^{15} + 3 x^{12} + 9 x^{5} + 9 x^{2} + 15$	$\He_3^2:C_6$ (as 18T455)	$4374$	$3$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^6 + 1,z^2 + 2$	undefined
3.1.18.37d2.74		$x^{18} + 3 x^{15} + 3 x^{12} + 9 x^{6} + 9 x^{5} + 9 x^{2} + 15$	$\He_3^2:C_6$ (as 18T455)	$4374$	$3$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^6 + 1,z^2 + 2$	undefined
3.1.18.37d2.75		$x^{18} + 3 x^{15} + 3 x^{12} + 18 x^{6} + 9 x^{5} + 9 x^{2} + 15$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^6 + 1,z^2 + 2$	undefined
3.1.18.37d2.76		$x^{18} + 3 x^{15} + 3 x^{12} + 9 x^{2} + 24$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^6 + 1,z^2 + 2$	undefined
3.1.18.37d2.77		$x^{18} + 3 x^{15} + 3 x^{12} + 9 x^{6} + 9 x^{2} + 24$	$\He_3^2:C_6$ (as 18T455)	$4374$	$3$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^6 + 1,z^2 + 2$	undefined
3.1.18.37d2.78		$x^{18} + 3 x^{15} + 3 x^{12} + 18 x^{6} + 9 x^{2} + 24$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^6 + 1,z^2 + 2$	undefined
3.1.18.37d2.79		$x^{18} + 3 x^{15} + 3 x^{12} + 9 x^{5} + 9 x^{2} + 24$	$\He_3^2:C_6$ (as 18T455)	$4374$	$3$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^6 + 1,z^2 + 2$	undefined
3.1.18.37d2.80		$x^{18} + 3 x^{15} + 3 x^{12} + 9 x^{6} + 9 x^{5} + 9 x^{2} + 24$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[20, 12, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^6 + 1,z^2 + 2$	undefined

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