LMFDB - $p$-adic family 3.1.9.18b

Defining polynomial

$x^{9} + 9 b_{13} x^{4} + 3 a_{3} x^{3} + 9 b_{11} x^{2} + 9 a_{10} x + 3$

Invariants

Residue field characteristic:	$3$
Degree:	$9$
Base field:	$\Q_{3}$
Ramification index $e$:	$9$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$18$
Artin slopes:	$[\frac{3}{2},\frac{5}{2}]$
Swan slopes:	$[\frac{1}{2},\frac{3}{2}]$
Means:	$\langle\frac{1}{3},\frac{10}{9}\rangle$
Rams:	$(\frac{1}{2},\frac{7}{2})$
Field count:	$36$ (complete)
Ambiguity:	$1$
Mass:	$36$
Absolute Mass:	$36$

Diagrams

Varying

Indices of inseparability:	$[10,3,0]$
Associated inertia:	$[1,1]$
Jump Set:	undefined

Galois groups and Hidden Artin slopes

Select desired size of Galois group.

		Galois groups of order 18
		$D_9$ (as 9T3)	$C_3:S_3$ (as 9T5)	Total
hidden slopes	$[\ ]_{2}$	1	1	2

		Galois groups of order 36
		$S_3^2$ (as 9T8)
hidden slopes	$[\ ]^{2}_{2}$	3

		Galois groups of order 54
		$C_9:C_6$ (as 9T10)	$C_3^2:C_6$ (as 9T11)	Total
hidden slopes	$[2]_{2}$	9	3	12

$[\ ]^{3}_{2}$	2	2	4

Total	11	5	16

		Galois groups of order 108
		$C_3^2:D_6$ (as 9T18)
hidden slopes	$[2]^{2}_{2}$	3

		Galois groups of order 324
		$C_3^3:D_6$ (as 9T24)
hidden slopes	$[\frac{3}{2},2]^{2}_{2}$	9

$[\frac{3}{2},\frac{5}{3}]^{2}_{2}$	3

Total	12

Fields

Galois group	Galois Artin	Indices

Assoc. Inertia	Jump Set	Fields per packet

Sort order	Select

Showing all 36

Download displayed columns for results to

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
3.1.9.18b1.1	9	$x^{9} + 3 x^{3} + 9 x + 3$	$(C_9:C_3):C_2$ (as 9T10)	$54$	$1$	$[\frac{3}{2}, 2, \frac{5}{2}]_{2}$	$[\frac{1}{2},1,\frac{3}{2}]_{2}$	$[2]_{2}$	$[1]_{2}$	$[10, 3, 0]$	$[1, 1]$	$z^3 + 1,z + 1$	undefined
3.1.9.18b1.2	9	$x^{9} + 9 x^{4} + 3 x^{3} + 9 x + 3$	$(C_9:C_3):C_2$ (as 9T10)	$54$	$1$	$[\frac{3}{2}, 2, \frac{5}{2}]_{2}$	$[\frac{1}{2},1,\frac{3}{2}]_{2}$	$[2]_{2}$	$[1]_{2}$	$[10, 3, 0]$	$[1, 1]$	$z^3 + 1,z + 1$	undefined
3.1.9.18b1.3	9	$x^{9} + 18 x^{4} + 3 x^{3} + 9 x + 3$	$(C_9:C_3):C_2$ (as 9T10)	$54$	$1$	$[\frac{3}{2}, 2, \frac{5}{2}]_{2}$	$[\frac{1}{2},1,\frac{3}{2}]_{2}$	$[2]_{2}$	$[1]_{2}$	$[10, 3, 0]$	$[1, 1]$	$z^3 + 1,z + 1$	undefined
3.1.9.18b1.4	1	$x^{9} + 3 x^{3} + 9 x^{2} + 9 x + 3$	$C_3^2:C_2$ (as 9T5)	$18$	$1$	$[\frac{3}{2}, \frac{5}{2}]_{2}$	$[\frac{1}{2},\frac{3}{2}]_{2}$	$[\ ]_{2}$	$[\ ]_{2}$	$[10, 3, 0]$	$[1, 1]$	$z^3 + 1,z + 1$	undefined
3.1.9.18b1.5	2	$x^{9} + 9 x^{4} + 3 x^{3} + 9 x^{2} + 9 x + 3$	$C_3^2 : C_6$ (as 9T11)	$54$	$1$	$[\frac{3}{2}, \frac{5}{2}]_{2}^{3}$	$[\frac{1}{2},\frac{3}{2}]_{2}^{3}$	$[\ ]^{3}_{2}$	$[\ ]^{3}_{2}$	$[10, 3, 0]$	$[1, 1]$	$z^3 + 1,z + 1$	undefined
3.1.9.18b1.6	2	$x^{9} + 18 x^{4} + 3 x^{3} + 9 x^{2} + 9 x + 3$	$C_3^2 : C_6$ (as 9T11)	$54$	$1$	$[\frac{3}{2}, \frac{5}{2}]_{2}^{3}$	$[\frac{1}{2},\frac{3}{2}]_{2}^{3}$	$[\ ]^{3}_{2}$	$[\ ]^{3}_{2}$	$[10, 3, 0]$	$[1, 1]$	$z^3 + 1,z + 1$	undefined
3.1.9.18b1.7	9	$x^{9} + 3 x^{3} + 18 x^{2} + 9 x + 3$	$(C_9:C_3):C_2$ (as 9T10)	$54$	$1$	$[\frac{3}{2}, 2, \frac{5}{2}]_{2}$	$[\frac{1}{2},1,\frac{3}{2}]_{2}$	$[2]_{2}$	$[1]_{2}$	$[10, 3, 0]$	$[1, 1]$	$z^3 + 1,z + 1$	undefined
3.1.9.18b1.8	9	$x^{9} + 9 x^{4} + 3 x^{3} + 18 x^{2} + 9 x + 3$	$(C_9:C_3):C_2$ (as 9T10)	$54$	$1$	$[\frac{3}{2}, 2, \frac{5}{2}]_{2}$	$[\frac{1}{2},1,\frac{3}{2}]_{2}$	$[2]_{2}$	$[1]_{2}$	$[10, 3, 0]$	$[1, 1]$	$z^3 + 1,z + 1$	undefined
3.1.9.18b1.9	9	$x^{9} + 18 x^{4} + 3 x^{3} + 18 x^{2} + 9 x + 3$	$(C_9:C_3):C_2$ (as 9T10)	$54$	$1$	$[\frac{3}{2}, 2, \frac{5}{2}]_{2}$	$[\frac{1}{2},1,\frac{3}{2}]_{2}$	$[2]_{2}$	$[1]_{2}$	$[10, 3, 0]$	$[1, 1]$	$z^3 + 1,z + 1$	undefined
3.1.9.18b1.10	9	$x^{9} + 6 x^{3} + 9 x + 3$	$(C_9:C_3):C_2$ (as 9T10)	$54$	$1$	$[\frac{3}{2}, 2, \frac{5}{2}]_{2}$	$[\frac{1}{2},1,\frac{3}{2}]_{2}$	$[2]_{2}$	$[1]_{2}$	$[10, 3, 0]$	$[1, 1]$	$z^3 + 1,z + 1$	undefined
3.1.9.18b1.11	9	$x^{9} + 9 x^{4} + 6 x^{3} + 9 x + 3$	$(C_9:C_3):C_2$ (as 9T10)	$54$	$1$	$[\frac{3}{2}, 2, \frac{5}{2}]_{2}$	$[\frac{1}{2},1,\frac{3}{2}]_{2}$	$[2]_{2}$	$[1]_{2}$	$[10, 3, 0]$	$[1, 1]$	$z^3 + 1,z + 1$	undefined
3.1.9.18b1.12	9	$x^{9} + 18 x^{4} + 6 x^{3} + 9 x + 3$	$(C_9:C_3):C_2$ (as 9T10)	$54$	$1$	$[\frac{3}{2}, 2, \frac{5}{2}]_{2}$	$[\frac{1}{2},1,\frac{3}{2}]_{2}$	$[2]_{2}$	$[1]_{2}$	$[10, 3, 0]$	$[1, 1]$	$z^3 + 1,z + 1$	undefined
3.1.9.18b1.13	3	$x^{9} + 6 x^{3} + 9 x^{2} + 9 x + 3$	$C_3^2 : C_6$ (as 9T11)	$54$	$1$	$[\frac{3}{2}, 2, \frac{5}{2}]_{2}$	$[\frac{1}{2},1,\frac{3}{2}]_{2}$	$[2]_{2}$	$[1]_{2}$	$[10, 3, 0]$	$[1, 1]$	$z^3 + 1,z + 1$	undefined
3.1.9.18b1.14	3	$x^{9} + 9 x^{4} + 6 x^{3} + 9 x^{2} + 9 x + 3$	$C_3^2 : C_6$ (as 9T11)	$54$	$1$	$[\frac{3}{2}, 2, \frac{5}{2}]_{2}$	$[\frac{1}{2},1,\frac{3}{2}]_{2}$	$[2]_{2}$	$[1]_{2}$	$[10, 3, 0]$	$[1, 1]$	$z^3 + 1,z + 1$	undefined
3.1.9.18b1.15	3	$x^{9} + 18 x^{4} + 6 x^{3} + 9 x^{2} + 9 x + 3$	$C_3^2 : C_6$ (as 9T11)	$54$	$1$	$[\frac{3}{2}, 2, \frac{5}{2}]_{2}$	$[\frac{1}{2},1,\frac{3}{2}]_{2}$	$[2]_{2}$	$[1]_{2}$	$[10, 3, 0]$	$[1, 1]$	$z^3 + 1,z + 1$	undefined
3.1.9.18b1.16	2	$x^{9} + 6 x^{3} + 18 x^{2} + 9 x + 3$	$(C_9:C_3):C_2$ (as 9T10)	$54$	$1$	$[\frac{3}{2}, \frac{5}{2}]_{2}^{3}$	$[\frac{1}{2},\frac{3}{2}]_{2}^{3}$	$[\ ]^{3}_{2}$	$[\ ]^{3}_{2}$	$[10, 3, 0]$	$[1, 1]$	$z^3 + 1,z + 1$	undefined
3.1.9.18b1.17	2	$x^{9} + 9 x^{4} + 6 x^{3} + 18 x^{2} + 9 x + 3$	$(C_9:C_3):C_2$ (as 9T10)	$54$	$1$	$[\frac{3}{2}, \frac{5}{2}]_{2}^{3}$	$[\frac{1}{2},\frac{3}{2}]_{2}^{3}$	$[\ ]^{3}_{2}$	$[\ ]^{3}_{2}$	$[10, 3, 0]$	$[1, 1]$	$z^3 + 1,z + 1$	undefined
3.1.9.18b1.18	1	$x^{9} + 18 x^{4} + 6 x^{3} + 18 x^{2} + 9 x + 3$	$D_{9}$ (as 9T3)	$18$	$1$	$[\frac{3}{2}, \frac{5}{2}]_{2}$	$[\frac{1}{2},\frac{3}{2}]_{2}$	$[\ ]_{2}$	$[\ ]_{2}$	$[10, 3, 0]$	$[1, 1]$	$z^3 + 1,z + 1$	undefined
3.1.9.18b2.1	9	$x^{9} + 3 x^{3} + 18 x + 3$	$((C_3^3:C_3):C_2):C_2$ (as 9T24)	$324$	$1$	$[\frac{3}{2}, \frac{3}{2}, 2, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},\frac{1}{2},1,\frac{3}{2}]_{2}^{2}$	$[\frac{3}{2},2]^{2}_{2}$	$[\frac{1}{2},1]^{2}_{2}$	$[10, 3, 0]$	$[1, 1]$	$z^3 + 1,z + 2$	undefined
3.1.9.18b2.2	9	$x^{9} + 9 x^{4} + 3 x^{3} + 18 x + 3$	$((C_3^3:C_3):C_2):C_2$ (as 9T24)	$324$	$1$	$[\frac{3}{2}, \frac{3}{2}, 2, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},\frac{1}{2},1,\frac{3}{2}]_{2}^{2}$	$[\frac{3}{2},2]^{2}_{2}$	$[\frac{1}{2},1]^{2}_{2}$	$[10, 3, 0]$	$[1, 1]$	$z^3 + 1,z + 2$	undefined
3.1.9.18b2.3	9	$x^{9} + 18 x^{4} + 3 x^{3} + 18 x + 3$	$((C_3^3:C_3):C_2):C_2$ (as 9T24)	$324$	$1$	$[\frac{3}{2}, \frac{3}{2}, 2, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},\frac{1}{2},1,\frac{3}{2}]_{2}^{2}$	$[\frac{3}{2},2]^{2}_{2}$	$[\frac{1}{2},1]^{2}_{2}$	$[10, 3, 0]$	$[1, 1]$	$z^3 + 1,z + 2$	undefined
3.1.9.18b2.4	3	$x^{9} + 3 x^{3} + 9 x^{2} + 18 x + 3$	$C_3^2 : D_{6} $ (as 9T18)	$108$	$1$	$[\frac{3}{2}, 2, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},1,\frac{3}{2}]_{2}^{2}$	$[2]^{2}_{2}$	$[1]^{2}_{2}$	$[10, 3, 0]$	$[1, 1]$	$z^3 + 1,z + 2$	undefined
3.1.9.18b2.5	3	$x^{9} + 9 x^{4} + 3 x^{3} + 9 x^{2} + 18 x + 3$	$C_3^2 : D_{6} $ (as 9T18)	$108$	$1$	$[\frac{3}{2}, 2, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},1,\frac{3}{2}]_{2}^{2}$	$[2]^{2}_{2}$	$[1]^{2}_{2}$	$[10, 3, 0]$	$[1, 1]$	$z^3 + 1,z + 2$	undefined
3.1.9.18b2.6	3	$x^{9} + 18 x^{4} + 3 x^{3} + 9 x^{2} + 18 x + 3$	$C_3^2 : D_{6} $ (as 9T18)	$108$	$1$	$[\frac{3}{2}, 2, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},1,\frac{3}{2}]_{2}^{2}$	$[2]^{2}_{2}$	$[1]^{2}_{2}$	$[10, 3, 0]$	$[1, 1]$	$z^3 + 1,z + 2$	undefined
3.1.9.18b2.7	3	$x^{9} + 3 x^{3} + 18 x^{2} + 18 x + 3$	$((C_3^3:C_3):C_2):C_2$ (as 9T24)	$324$	$1$	$[\frac{3}{2}, \frac{3}{2}, \frac{5}{3}, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},\frac{1}{2},\frac{2}{3},\frac{3}{2}]_{2}^{2}$	$[\frac{3}{2},\frac{5}{3}]^{2}_{2}$	$[\frac{1}{2},\frac{2}{3}]^{2}_{2}$	$[10, 3, 0]$	$[1, 1]$	$z^3 + 1,z + 2$	undefined
3.1.9.18b2.8	3	$x^{9} + 9 x^{4} + 3 x^{3} + 18 x^{2} + 18 x + 3$	$((C_3^3:C_3):C_2):C_2$ (as 9T24)	$324$	$1$	$[\frac{3}{2}, \frac{3}{2}, \frac{5}{3}, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},\frac{1}{2},\frac{2}{3},\frac{3}{2}]_{2}^{2}$	$[\frac{3}{2},\frac{5}{3}]^{2}_{2}$	$[\frac{1}{2},\frac{2}{3}]^{2}_{2}$	$[10, 3, 0]$	$[1, 1]$	$z^3 + 1,z + 2$	undefined
3.1.9.18b2.9	3	$x^{9} + 18 x^{4} + 3 x^{3} + 18 x^{2} + 18 x + 3$	$((C_3^3:C_3):C_2):C_2$ (as 9T24)	$324$	$1$	$[\frac{3}{2}, \frac{3}{2}, \frac{5}{3}, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},\frac{1}{2},\frac{2}{3},\frac{3}{2}]_{2}^{2}$	$[\frac{3}{2},\frac{5}{3}]^{2}_{2}$	$[\frac{1}{2},\frac{2}{3}]^{2}_{2}$	$[10, 3, 0]$	$[1, 1]$	$z^3 + 1,z + 2$	undefined
3.1.9.18b2.10	9	$x^{9} + 6 x^{3} + 18 x + 3$	$((C_3^3:C_3):C_2):C_2$ (as 9T24)	$324$	$1$	$[\frac{3}{2}, \frac{3}{2}, 2, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},\frac{1}{2},1,\frac{3}{2}]_{2}^{2}$	$[\frac{3}{2},2]^{2}_{2}$	$[\frac{1}{2},1]^{2}_{2}$	$[10, 3, 0]$	$[1, 1]$	$z^3 + 1,z + 2$	undefined
3.1.9.18b2.11	9	$x^{9} + 9 x^{4} + 6 x^{3} + 18 x + 3$	$((C_3^3:C_3):C_2):C_2$ (as 9T24)	$324$	$1$	$[\frac{3}{2}, \frac{3}{2}, 2, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},\frac{1}{2},1,\frac{3}{2}]_{2}^{2}$	$[\frac{3}{2},2]^{2}_{2}$	$[\frac{1}{2},1]^{2}_{2}$	$[10, 3, 0]$	$[1, 1]$	$z^3 + 1,z + 2$	undefined
3.1.9.18b2.12	9	$x^{9} + 18 x^{4} + 6 x^{3} + 18 x + 3$	$((C_3^3:C_3):C_2):C_2$ (as 9T24)	$324$	$1$	$[\frac{3}{2}, \frac{3}{2}, 2, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},\frac{1}{2},1,\frac{3}{2}]_{2}^{2}$	$[\frac{3}{2},2]^{2}_{2}$	$[\frac{1}{2},1]^{2}_{2}$	$[10, 3, 0]$	$[1, 1]$	$z^3 + 1,z + 2$	undefined
3.1.9.18b2.13	3	$x^{9} + 6 x^{3} + 9 x^{2} + 18 x + 3$	$S_3^2$ (as 9T8)	$36$	$1$	$[\frac{3}{2}, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},\frac{3}{2}]_{2}^{2}$	$[\ ]^{2}_{2}$	$[\ ]^{2}_{2}$	$[10, 3, 0]$	$[1, 1]$	$z^3 + 1,z + 2$	undefined
3.1.9.18b2.14	3	$x^{9} + 9 x^{4} + 6 x^{3} + 9 x^{2} + 18 x + 3$	$S_3^2$ (as 9T8)	$36$	$1$	$[\frac{3}{2}, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},\frac{3}{2}]_{2}^{2}$	$[\ ]^{2}_{2}$	$[\ ]^{2}_{2}$	$[10, 3, 0]$	$[1, 1]$	$z^3 + 1,z + 2$	undefined
3.1.9.18b2.15	3	$x^{9} + 18 x^{4} + 6 x^{3} + 9 x^{2} + 18 x + 3$	$S_3^2$ (as 9T8)	$36$	$1$	$[\frac{3}{2}, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},\frac{3}{2}]_{2}^{2}$	$[\ ]^{2}_{2}$	$[\ ]^{2}_{2}$	$[10, 3, 0]$	$[1, 1]$	$z^3 + 1,z + 2$	undefined
3.1.9.18b2.16	9	$x^{9} + 6 x^{3} + 18 x^{2} + 18 x + 3$	$((C_3^3:C_3):C_2):C_2$ (as 9T24)	$324$	$1$	$[\frac{3}{2}, \frac{3}{2}, 2, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},\frac{1}{2},1,\frac{3}{2}]_{2}^{2}$	$[\frac{3}{2},2]^{2}_{2}$	$[\frac{1}{2},1]^{2}_{2}$	$[10, 3, 0]$	$[1, 1]$	$z^3 + 1,z + 2$	undefined
3.1.9.18b2.17	9	$x^{9} + 9 x^{4} + 6 x^{3} + 18 x^{2} + 18 x + 3$	$((C_3^3:C_3):C_2):C_2$ (as 9T24)	$324$	$1$	$[\frac{3}{2}, \frac{3}{2}, 2, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},\frac{1}{2},1,\frac{3}{2}]_{2}^{2}$	$[\frac{3}{2},2]^{2}_{2}$	$[\frac{1}{2},1]^{2}_{2}$	$[10, 3, 0]$	$[1, 1]$	$z^3 + 1,z + 2$	undefined
3.1.9.18b2.18	9	$x^{9} + 18 x^{4} + 6 x^{3} + 18 x^{2} + 18 x + 3$	$((C_3^3:C_3):C_2):C_2$ (as 9T24)	$324$	$1$	$[\frac{3}{2}, \frac{3}{2}, 2, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},\frac{1}{2},1,\frac{3}{2}]_{2}^{2}$	$[\frac{3}{2},2]^{2}_{2}$	$[\frac{1}{2},1]^{2}_{2}$	$[10, 3, 0]$	$[1, 1]$	$z^3 + 1,z + 2$	undefined

Download displayed columns for results to

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Label	3.1.9.18b
Base	3.1.1.0a1.1
Degree	\(9\)
e	\(9\)
f	\(1\)
c	\(18\)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Defining polynomial

Invariants

Diagrams

Varying

Galois groups and Hidden Artin slopes

Fields

Showing all 36

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