LMFDB - $p$-adic family 3.1.18.40d

Defining polynomial

$x^{18} + 3 a_{15} x^{15} + 9 c_{27} x^{9} + 9 b_{26} x^{8} + 9 b_{25} x^{7} + 9 a_{23} x^{5} + 9 b_{21} x^{3} + 3 d_{0}$

Invariants

Residue field characteristic:	$3$
Degree:	$18$
Base field:	$\Q_{3}$
Ramification index $e$:	$18$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$40$
Artin slopes:	$[\frac{9}{4},\frac{5}{2}]$
Swan slopes:	$[\frac{5}{4},\frac{3}{2}]$
Means:	$\langle\frac{5}{6},\frac{23}{18}\rangle$
Rams:	$(\frac{5}{2},4)$
Field count:	$216$ (complete)
Ambiguity:	$6$
Mass:	$108$
Absolute Mass:	$108$

Diagrams

Varying

Indices of inseparability:	$[23,15,0]$
Associated inertia:	$[1,1,1]$ (show 162), $[1,1,2]$ (show 54)
Jump Set:	undefined (show 108), $[1,7,25]$ (show 108)

Fields

Galois group	Galois Artin	Indices

Assoc. Inertia	Jump Set

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Showing 1-50 of 54

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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
3.1.18.40d1.1	$x^{18} + 3 x^{15} + 18 x^{5} + 9 x^{3} + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 1,z^2 + 1$	$[1, 7, 25]$
3.1.18.40d1.2	$x^{18} + 3 x^{15} + 9 x^{8} + 18 x^{5} + 9 x^{3} + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 1,z^2 + 1$	$[1, 7, 25]$
3.1.18.40d1.3	$x^{18} + 3 x^{15} + 18 x^{8} + 18 x^{5} + 9 x^{3} + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 1,z^2 + 1$	$[1, 7, 25]$
3.1.18.40d1.4	$x^{18} + 3 x^{15} + 9 x^{7} + 18 x^{5} + 9 x^{3} + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 1,z^2 + 1$	$[1, 7, 25]$
3.1.18.40d1.5	$x^{18} + 3 x^{15} + 9 x^{8} + 9 x^{7} + 18 x^{5} + 9 x^{3} + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 1,z^2 + 1$	$[1, 7, 25]$
3.1.18.40d1.6	$x^{18} + 3 x^{15} + 18 x^{8} + 9 x^{7} + 18 x^{5} + 9 x^{3} + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 1,z^2 + 1$	$[1, 7, 25]$
3.1.18.40d1.7	$x^{18} + 3 x^{15} + 18 x^{7} + 18 x^{5} + 9 x^{3} + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 1,z^2 + 1$	$[1, 7, 25]$
3.1.18.40d1.8	$x^{18} + 3 x^{15} + 9 x^{8} + 18 x^{7} + 18 x^{5} + 9 x^{3} + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 1,z^2 + 1$	$[1, 7, 25]$
3.1.18.40d1.9	$x^{18} + 3 x^{15} + 18 x^{8} + 18 x^{7} + 18 x^{5} + 9 x^{3} + 3$	$\He_3^2:D_4$ (as 18T509)	$5832$	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 1,z^2 + 1$	$[1, 7, 25]$
3.1.18.40d1.10	$x^{18} + 6 x^{15} + 9 x^{5} + 3$	$\He_3^2:D_4$ (as 18T509)	$5832$	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 2,2 z^2 + 2$	$[1, 7, 25]$
3.1.18.40d1.11	$x^{18} + 6 x^{15} + 9 x^{8} + 9 x^{5} + 3$	$\He_3^2:D_4$ (as 18T509)	$5832$	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 2,2 z^2 + 2$	$[1, 7, 25]$
3.1.18.40d1.12	$x^{18} + 6 x^{15} + 18 x^{8} + 9 x^{5} + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 2,2 z^2 + 2$	$[1, 7, 25]$
3.1.18.40d1.13	$x^{18} + 6 x^{15} + 9 x^{7} + 9 x^{5} + 3$	$\He_3^2:D_4$ (as 18T509)	$5832$	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 2,2 z^2 + 2$	$[1, 7, 25]$
3.1.18.40d1.14	$x^{18} + 6 x^{15} + 9 x^{8} + 9 x^{7} + 9 x^{5} + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 2,2 z^2 + 2$	$[1, 7, 25]$
3.1.18.40d1.15	$x^{18} + 6 x^{15} + 18 x^{8} + 9 x^{7} + 9 x^{5} + 3$	$\He_3^2:D_4$ (as 18T509)	$5832$	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 2,2 z^2 + 2$	$[1, 7, 25]$
3.1.18.40d1.16	$x^{18} + 6 x^{15} + 18 x^{7} + 9 x^{5} + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 2,2 z^2 + 2$	$[1, 7, 25]$
3.1.18.40d1.17	$x^{18} + 6 x^{15} + 9 x^{8} + 18 x^{7} + 9 x^{5} + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 2,2 z^2 + 2$	$[1, 7, 25]$
3.1.18.40d1.18	$x^{18} + 6 x^{15} + 18 x^{8} + 18 x^{7} + 9 x^{5} + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 2,2 z^2 + 2$	$[1, 7, 25]$
3.1.18.40d1.19	$x^{18} + 6 x^{15} + 9 x^{5} + 9 x^{3} + 3$	$\He_3^2:D_4$ (as 18T509)	$5832$	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 2,2 z^2 + 2$	$[1, 7, 25]$
3.1.18.40d1.20	$x^{18} + 6 x^{15} + 9 x^{8} + 9 x^{5} + 9 x^{3} + 3$	$\He_3^2:D_4$ (as 18T509)	$5832$	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 2,2 z^2 + 2$	$[1, 7, 25]$
3.1.18.40d1.21	$x^{18} + 6 x^{15} + 18 x^{8} + 9 x^{5} + 9 x^{3} + 3$	$\He_3^2:D_4$ (as 18T509)	$5832$	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 2,2 z^2 + 2$	$[1, 7, 25]$
3.1.18.40d1.22	$x^{18} + 6 x^{15} + 9 x^{7} + 9 x^{5} + 9 x^{3} + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 2,2 z^2 + 2$	$[1, 7, 25]$
3.1.18.40d1.23	$x^{18} + 6 x^{15} + 9 x^{8} + 9 x^{7} + 9 x^{5} + 9 x^{3} + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 2,2 z^2 + 2$	$[1, 7, 25]$
3.1.18.40d1.24	$x^{18} + 6 x^{15} + 18 x^{8} + 9 x^{7} + 9 x^{5} + 9 x^{3} + 3$	$\He_3^2:D_4$ (as 18T509)	$5832$	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 2,2 z^2 + 2$	$[1, 7, 25]$
3.1.18.40d1.25	$x^{18} + 6 x^{15} + 18 x^{7} + 9 x^{5} + 9 x^{3} + 3$	$\He_3^2:D_4$ (as 18T509)	$5832$	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 2,2 z^2 + 2$	$[1, 7, 25]$
3.1.18.40d1.26	$x^{18} + 6 x^{15} + 9 x^{8} + 18 x^{7} + 9 x^{5} + 9 x^{3} + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 2,2 z^2 + 2$	$[1, 7, 25]$
3.1.18.40d1.27	$x^{18} + 6 x^{15} + 18 x^{8} + 18 x^{7} + 9 x^{5} + 9 x^{3} + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 2,2 z^2 + 2$	$[1, 7, 25]$
3.1.18.40d1.28	$x^{18} + 3 x^{15} + 9 x^{5} + 6$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 2,2 z^2 + 2$	undefined
3.1.18.40d1.29	$x^{18} + 3 x^{15} + 9 x^{8} + 9 x^{5} + 6$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 2,2 z^2 + 2$	undefined
3.1.18.40d1.30	$x^{18} + 3 x^{15} + 18 x^{8} + 9 x^{5} + 6$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 2,2 z^2 + 2$	undefined
3.1.18.40d1.31	$x^{18} + 3 x^{15} + 9 x^{7} + 9 x^{5} + 6$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 2,2 z^2 + 2$	undefined
3.1.18.40d1.32	$x^{18} + 3 x^{15} + 9 x^{8} + 9 x^{7} + 9 x^{5} + 6$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 2,2 z^2 + 2$	undefined
3.1.18.40d1.33	$x^{18} + 3 x^{15} + 18 x^{8} + 9 x^{7} + 9 x^{5} + 6$	$C_3^3:D_{12}$ (as 18T210)	$648$	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 2,2 z^2 + 2$	undefined
3.1.18.40d1.34	$x^{18} + 3 x^{15} + 18 x^{7} + 9 x^{5} + 6$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 2,2 z^2 + 2$	undefined
3.1.18.40d1.35	$x^{18} + 3 x^{15} + 9 x^{8} + 18 x^{7} + 9 x^{5} + 6$	$C_3^3:D_{12}$ (as 18T210)	$648$	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 2,2 z^2 + 2$	undefined
3.1.18.40d1.36	$x^{18} + 3 x^{15} + 18 x^{8} + 18 x^{7} + 9 x^{5} + 6$	$C_3^3:D_{12}$ (as 18T210)	$648$	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 2,2 z^2 + 2$	undefined
3.1.18.40d1.37	$x^{18} + 3 x^{15} + 9 x^{5} + 9 x^{3} + 6$	$C_3^3:D_{12}$ (as 18T210)	$648$	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 2,2 z^2 + 2$	undefined
3.1.18.40d1.38	$x^{18} + 3 x^{15} + 9 x^{8} + 9 x^{5} + 9 x^{3} + 6$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 2,2 z^2 + 2$	undefined
3.1.18.40d1.39	$x^{18} + 3 x^{15} + 18 x^{8} + 9 x^{5} + 9 x^{3} + 6$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 2,2 z^2 + 2$	undefined
3.1.18.40d1.40	$x^{18} + 3 x^{15} + 9 x^{7} + 9 x^{5} + 9 x^{3} + 6$	$S_3^2:S_3$ (as 18T106)	$216$	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 2,2 z^2 + 2$	undefined
3.1.18.40d1.41	$x^{18} + 3 x^{15} + 9 x^{8} + 9 x^{7} + 9 x^{5} + 9 x^{3} + 6$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 2,2 z^2 + 2$	undefined
3.1.18.40d1.42	$x^{18} + 3 x^{15} + 18 x^{8} + 9 x^{7} + 9 x^{5} + 9 x^{3} + 6$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 2,2 z^2 + 2$	undefined
3.1.18.40d1.43	$x^{18} + 3 x^{15} + 18 x^{7} + 9 x^{5} + 9 x^{3} + 6$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 2,2 z^2 + 2$	undefined
3.1.18.40d1.44	$x^{18} + 3 x^{15} + 9 x^{8} + 18 x^{7} + 9 x^{5} + 9 x^{3} + 6$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 2,2 z^2 + 2$	undefined
3.1.18.40d1.45	$x^{18} + 3 x^{15} + 18 x^{8} + 18 x^{7} + 9 x^{5} + 9 x^{3} + 6$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 2,2 z^2 + 2$	undefined
3.1.18.40d1.46	$x^{18} + 6 x^{15} + 18 x^{5} + 9 x^{3} + 6$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 1,z^2 + 1$	undefined
3.1.18.40d1.47	$x^{18} + 6 x^{15} + 9 x^{8} + 18 x^{5} + 9 x^{3} + 6$	$C_3^3:D_{12}$ (as 18T210)	$648$	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 1,z^2 + 1$	undefined
3.1.18.40d1.48	$x^{18} + 6 x^{15} + 18 x^{8} + 18 x^{5} + 9 x^{3} + 6$	$C_3^3:D_{12}$ (as 18T210)	$648$	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 1,z^2 + 1$	undefined
3.1.18.40d1.49	$x^{18} + 6 x^{15} + 9 x^{7} + 18 x^{5} + 9 x^{3} + 6$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 1,z^2 + 1$	undefined
3.1.18.40d1.50	$x^{18} + 6 x^{15} + 9 x^{8} + 9 x^{7} + 18 x^{5} + 9 x^{3} + 6$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 2]$	$z^9 + 2,2 z^3 + 1,z^2 + 1$	undefined

Next 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	3.1.18.40d
Base	3.1.1.0a1.1
Degree	\(18\)
e	\(18\)
f	\(1\)
c	\(40\)

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