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

Sort order	Select

Showing 1-50 of 162

Next Download displayed columns for results to

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
3.1.18.40d2.1	$x^{18} + 3 x^{15} + 9 x^{5} + 3$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.2	$x^{18} + 3 x^{15} + 9 x^{9} + 9 x^{5} + 3$	$C_3\wr D_4$ (as 18T189)	$648$	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.3	$x^{18} + 3 x^{15} + 18 x^{9} + 9 x^{5} + 3$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.4	$x^{18} + 3 x^{15} + 9 x^{8} + 9 x^{5} + 3$	$S_3^2:S_3$ (as 18T103)	$216$	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.5	$x^{18} + 3 x^{15} + 9 x^{9} + 9 x^{8} + 9 x^{5} + 3$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.6	$x^{18} + 3 x^{15} + 18 x^{9} + 9 x^{8} + 9 x^{5} + 3$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.7	$x^{18} + 3 x^{15} + 18 x^{8} + 9 x^{5} + 3$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.8	$x^{18} + 3 x^{15} + 9 x^{9} + 18 x^{8} + 9 x^{5} + 3$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.9	$x^{18} + 3 x^{15} + 18 x^{9} + 18 x^{8} + 9 x^{5} + 3$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.10	$x^{18} + 3 x^{15} + 9 x^{7} + 9 x^{5} + 3$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.11	$x^{18} + 3 x^{15} + 9 x^{9} + 9 x^{7} + 9 x^{5} + 3$	$C_3\wr D_4$ (as 18T189)	$648$	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.12	$x^{18} + 3 x^{15} + 18 x^{9} + 9 x^{7} + 9 x^{5} + 3$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.13	$x^{18} + 3 x^{15} + 9 x^{8} + 9 x^{7} + 9 x^{5} + 3$	$C_3\wr D_4$ (as 18T189)	$648$	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.14	$x^{18} + 3 x^{15} + 9 x^{9} + 9 x^{8} + 9 x^{7} + 9 x^{5} + 3$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.15	$x^{18} + 3 x^{15} + 18 x^{9} + 9 x^{8} + 9 x^{7} + 9 x^{5} + 3$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.16	$x^{18} + 3 x^{15} + 18 x^{8} + 9 x^{7} + 9 x^{5} + 3$	$C_3\wr D_4$ (as 18T189)	$648$	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.17	$x^{18} + 3 x^{15} + 9 x^{9} + 18 x^{8} + 9 x^{7} + 9 x^{5} + 3$	$C_3\wr D_4$ (as 18T189)	$648$	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.18	$x^{18} + 3 x^{15} + 18 x^{9} + 18 x^{8} + 9 x^{7} + 9 x^{5} + 3$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.19	$x^{18} + 3 x^{15} + 18 x^{7} + 9 x^{5} + 3$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.20	$x^{18} + 3 x^{15} + 9 x^{9} + 18 x^{7} + 9 x^{5} + 3$	$C_3\wr D_4$ (as 18T189)	$648$	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.21	$x^{18} + 3 x^{15} + 18 x^{9} + 18 x^{7} + 9 x^{5} + 3$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.22	$x^{18} + 3 x^{15} + 9 x^{8} + 18 x^{7} + 9 x^{5} + 3$	$C_3\wr D_4$ (as 18T189)	$648$	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.23	$x^{18} + 3 x^{15} + 9 x^{9} + 9 x^{8} + 18 x^{7} + 9 x^{5} + 3$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.24	$x^{18} + 3 x^{15} + 18 x^{9} + 9 x^{8} + 18 x^{7} + 9 x^{5} + 3$	$C_3\wr D_4$ (as 18T189)	$648$	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.25	$x^{18} + 3 x^{15} + 18 x^{8} + 18 x^{7} + 9 x^{5} + 3$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.26	$x^{18} + 3 x^{15} + 9 x^{9} + 18 x^{8} + 18 x^{7} + 9 x^{5} + 3$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.27	$x^{18} + 3 x^{15} + 18 x^{9} + 18 x^{8} + 18 x^{7} + 9 x^{5} + 3$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.28	$x^{18} + 3 x^{15} + 9 x^{5} + 9 x^{3} + 3$	$C_3\wr D_4$ (as 18T189)	$648$	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.29	$x^{18} + 3 x^{15} + 9 x^{9} + 9 x^{5} + 9 x^{3} + 3$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.30	$x^{18} + 3 x^{15} + 18 x^{9} + 9 x^{5} + 9 x^{3} + 3$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.31	$x^{18} + 3 x^{15} + 9 x^{8} + 9 x^{5} + 9 x^{3} + 3$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.32	$x^{18} + 3 x^{15} + 9 x^{9} + 9 x^{8} + 9 x^{5} + 9 x^{3} + 3$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.33	$x^{18} + 3 x^{15} + 18 x^{9} + 9 x^{8} + 9 x^{5} + 9 x^{3} + 3$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.34	$x^{18} + 3 x^{15} + 18 x^{8} + 9 x^{5} + 9 x^{3} + 3$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.35	$x^{18} + 3 x^{15} + 9 x^{9} + 18 x^{8} + 9 x^{5} + 9 x^{3} + 3$	$C_3\wr D_4$ (as 18T189)	$648$	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.36	$x^{18} + 3 x^{15} + 18 x^{9} + 18 x^{8} + 9 x^{5} + 9 x^{3} + 3$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.37	$x^{18} + 3 x^{15} + 9 x^{7} + 9 x^{5} + 9 x^{3} + 3$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.38	$x^{18} + 3 x^{15} + 9 x^{9} + 9 x^{7} + 9 x^{5} + 9 x^{3} + 3$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.39	$x^{18} + 3 x^{15} + 18 x^{9} + 9 x^{7} + 9 x^{5} + 9 x^{3} + 3$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.40	$x^{18} + 3 x^{15} + 9 x^{8} + 9 x^{7} + 9 x^{5} + 9 x^{3} + 3$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.41	$x^{18} + 3 x^{15} + 9 x^{9} + 9 x^{8} + 9 x^{7} + 9 x^{5} + 9 x^{3} + 3$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.42	$x^{18} + 3 x^{15} + 18 x^{9} + 9 x^{8} + 9 x^{7} + 9 x^{5} + 9 x^{3} + 3$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.43	$x^{18} + 3 x^{15} + 18 x^{8} + 9 x^{7} + 9 x^{5} + 9 x^{3} + 3$	$C_3\wr D_4$ (as 18T189)	$648$	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.44	$x^{18} + 3 x^{15} + 9 x^{9} + 18 x^{8} + 9 x^{7} + 9 x^{5} + 9 x^{3} + 3$	$C_3\wr D_4$ (as 18T189)	$648$	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.45	$x^{18} + 3 x^{15} + 18 x^{9} + 18 x^{8} + 9 x^{7} + 9 x^{5} + 9 x^{3} + 3$	$C_3\wr D_4$ (as 18T189)	$648$	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.46	$x^{18} + 3 x^{15} + 18 x^{7} + 9 x^{5} + 9 x^{3} + 3$	$S_3^2:S_3$ (as 18T103)	$216$	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.47	$x^{18} + 3 x^{15} + 9 x^{9} + 18 x^{7} + 9 x^{5} + 9 x^{3} + 3$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.48	$x^{18} + 3 x^{15} + 18 x^{9} + 18 x^{7} + 9 x^{5} + 9 x^{3} + 3$	$C_3\wr D_4$ (as 18T189)	$648$	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.49	$x^{18} + 3 x^{15} + 9 x^{8} + 18 x^{7} + 9 x^{5} + 9 x^{3} + 3$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$
3.1.18.40d2.50	$x^{18} + 3 x^{15} + 9 x^{9} + 9 x^{8} + 18 x^{7} + 9 x^{5} + 9 x^{3} + 3$	$C_3\wr D_4$ (as 18T189)	$648$	$3$	not computed	not computed	not computed	not computed	$[23, 15, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 1,z^2 + 2$	$[1, 7, 25]$

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	\(40\)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Defining polynomial

Invariants

Diagrams

Varying

Fields

Showing 1-50 of 162

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