LMFDB - $p$-adic family 3.2.6.22a

Defining polynomial over unramified subextension

$x^{6} + 9 c_{9} x^{3} + 9 b_{8} x^{2} + 9 b_{7} x + 3 d_{0}$

Invariants

Residue field characteristic:	$3$
Degree:	$12$
Base field:	$\Q_{3}$
Ramification index $e$:	$6$
Residue field degree $f$:	$2$
Discriminant exponent $c$:	$22$
Artin slopes:	$[\frac{5}{2}]$
Swan slopes:	$[\frac{3}{2}]$
Means:	$\langle1\rangle$
Rams:	$(3)$
Field count:	$96$ (complete)
Ambiguity:	$12$
Mass:	$81$
Absolute Mass:	$81/2$

Diagrams

Varying

Indices of inseparability:	$[6,0]$
Associated inertia:	$[1,1]$ (show 72), $[1,2]$ (show 24)
Jump Set:	undefined (show 24), $[1,7]$ (show 72)

Galois groups and Hidden Artin slopes

Select desired size of Galois group.

		Galois groups of order 12
		$D_6$ (as 12T3)
hidden slopes	$[\ ]$	3

		Galois groups of order 24
		$C_4\times S_3$ (as 12T11)
hidden slopes	$[\ ]^{2}$	3

		Galois groups of order 36
		$S_3^2$ (as 12T16)	$C_6\times S_3$ (as 12T18)	Total
hidden slopes	$[2]$	3	9	12

$[\frac{3}{2}]$	3		3

$[\ ]^{3}$		3	3

Total	6	12	18

		Galois groups of order 72
		$C_6.D_6$ (as 12T39)
hidden slopes	$[\frac{3}{2}]^{2}$	3

		Galois groups of order 108
		$C_3\times S_3^2$ (as 12T70)	$C_3:S_3^2$ (as 12T71)	Total
hidden slopes	$[2,2]$	9		9

$[2]^{3}$	3		3

$[\frac{3}{2},2]$	9	3	12

$[\frac{3}{2}]^{3}$	3		3

Total	24	3	27

		Galois groups of order 216
		$C_3^2:C_4\times S_3$ (as 12T119)
hidden slopes	$[2,2]^{2}$	6

		Galois groups of order 324
		$C_3\wr C_2^2$ (as 12T130)
hidden slopes	$[\frac{3}{2},2,2]$	18

$[\frac{3}{2},2]^{3}$	6

Total	24

		Galois groups of order 648
		$C_3^3:(C_4\times S_3)$ (as 12T170)
hidden slopes	$[\frac{3}{2},2,2]^{2}$	12

Fields

Galois group	Galois Artin	Indices

Assoc. Inertia	Jump Set	Fields per packet

Sort order	Select

Showing 1-50 of 96

Next 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
3.2.6.22a1.1	3	$( x^{2} + 2 x + 2 )^{6} + 3$	$D_6$ (as 12T3)	$12$	$12$	$[\frac{5}{2}]_{2}^{2}$	$[\frac{3}{2}]_{2}^{2}$	$[\ ]$	$[\ ]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.2	3	$( x^{2} + 2 x + 2 )^{6} + 9 ( x^{2} + 2 x + 2 )^{3} + 3$	$C_6\times S_3$ (as 12T18)	$36$	$6$	$[\frac{5}{2}]_{2}^{6}$	$[\frac{3}{2}]_{2}^{6}$	$[\ ]^{3}$	$[\ ]^{3}$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.3	3	$( x^{2} + 2 x + 2 )^{6} + 9 x ( x^{2} + 2 x + 2 )^{2} + 3$	$S_3^2$ (as 12T16)	$36$	$6$	$[\frac{3}{2}, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},\frac{3}{2}]_{2}^{2}$	$[\frac{3}{2}]$	$[\frac{1}{2}]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.4	3	$( x^{2} + 2 x + 2 )^{6} + 9 ( x^{2} + 2 x + 2 )^{3} + 9 x ( x^{2} + 2 x + 2 )^{2} + 3$	$C_3\times S_3^2$ (as 12T70)	$108$	$3$	$[\frac{3}{2}, \frac{5}{2}]_{2}^{6}$	$[\frac{1}{2},\frac{3}{2}]_{2}^{6}$	$[\frac{3}{2}]^{3}$	$[\frac{1}{2}]^{3}$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.5	3	$( x^{2} + 2 x + 2 )^{6} + 9 ( x^{2} + 2 x + 2 )^{2} + 3$	$D_6$ (as 12T3)	$12$	$12$	$[\frac{5}{2}]_{2}^{2}$	$[\frac{3}{2}]_{2}^{2}$	$[\ ]$	$[\ ]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.6	3	$( x^{2} + 2 x + 2 )^{6} + 9 ( x^{2} + 2 x + 2 )^{3} + 9 ( x^{2} + 2 x + 2 )^{2} + 3$	$C_6\times S_3$ (as 12T18)	$36$	$6$	$[\frac{5}{2}]_{2}^{6}$	$[\frac{3}{2}]_{2}^{6}$	$[\ ]^{3}$	$[\ ]^{3}$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.7	3	$( x^{2} + 2 x + 2 )^{6} + \left(9 x + 9\right) ( x^{2} + 2 x + 2 )^{2} + 3$	$S_3^2$ (as 12T16)	$36$	$6$	$[\frac{3}{2}, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},\frac{3}{2}]_{2}^{2}$	$[\frac{3}{2}]$	$[\frac{1}{2}]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.8	3	$( x^{2} + 2 x + 2 )^{6} + 9 ( x^{2} + 2 x + 2 )^{3} + \left(9 x + 9\right) ( x^{2} + 2 x + 2 )^{2} + 3$	$C_3\times S_3^2$ (as 12T70)	$108$	$3$	$[\frac{3}{2}, \frac{5}{2}]_{2}^{6}$	$[\frac{1}{2},\frac{3}{2}]_{2}^{6}$	$[\frac{3}{2}]^{3}$	$[\frac{1}{2}]^{3}$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.9	3	$( x^{2} + 2 x + 2 )^{6} + 18 ( x^{2} + 2 x + 2 )^{2} + 3$	$D_6$ (as 12T3)	$12$	$12$	$[\frac{5}{2}]_{2}^{2}$	$[\frac{3}{2}]_{2}^{2}$	$[\ ]$	$[\ ]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.10	3	$( x^{2} + 2 x + 2 )^{6} + 9 ( x^{2} + 2 x + 2 )^{3} + 18 ( x^{2} + 2 x + 2 )^{2} + 3$	$C_6\times S_3$ (as 12T18)	$36$	$6$	$[\frac{5}{2}]_{2}^{6}$	$[\frac{3}{2}]_{2}^{6}$	$[\ ]^{3}$	$[\ ]^{3}$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.11	3	$( x^{2} + 2 x + 2 )^{6} + \left(9 x + 18\right) ( x^{2} + 2 x + 2 )^{2} + 3$	$S_3^2$ (as 12T16)	$36$	$6$	$[\frac{3}{2}, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},\frac{3}{2}]_{2}^{2}$	$[\frac{3}{2}]$	$[\frac{1}{2}]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.12	3	$( x^{2} + 2 x + 2 )^{6} + 9 ( x^{2} + 2 x + 2 )^{3} + \left(9 x + 18\right) ( x^{2} + 2 x + 2 )^{2} + 3$	$C_3\times S_3^2$ (as 12T70)	$108$	$3$	$[\frac{3}{2}, \frac{5}{2}]_{2}^{6}$	$[\frac{1}{2},\frac{3}{2}]_{2}^{6}$	$[\frac{3}{2}]^{3}$	$[\frac{1}{2}]^{3}$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.13	18	$( x^{2} + 2 x + 2 )^{6} + 9 x ( x^{2} + 2 x + 2 ) + 3$	$C_3\wr C_2^2$ (as 12T130)	$324$	$3$	$[\frac{3}{2}, 2, 2, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},1,1,\frac{3}{2}]_{2}^{2}$	$[\frac{3}{2},2,2]$	$[\frac{1}{2},1,1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.14	18	$( x^{2} + 2 x + 2 )^{6} + 9 ( x^{2} + 2 x + 2 )^{3} + 9 x ( x^{2} + 2 x + 2 ) + 3$	$C_3\wr C_2^2$ (as 12T130)	$324$	$3$	$[\frac{3}{2}, 2, 2, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},1,1,\frac{3}{2}]_{2}^{2}$	$[\frac{3}{2},2,2]$	$[\frac{1}{2},1,1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.15	18	$( x^{2} + 2 x + 2 )^{6} + 18 ( x^{2} + 2 x + 2 )^{3} + 9 x ( x^{2} + 2 x + 2 ) + 3$	$C_3\wr C_2^2$ (as 12T130)	$324$	$3$	$[\frac{3}{2}, 2, 2, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},1,1,\frac{3}{2}]_{2}^{2}$	$[\frac{3}{2},2,2]$	$[\frac{1}{2},1,1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.16	18	$( x^{2} + 2 x + 2 )^{6} + 9 x ( x^{2} + 2 x + 2 )^{2} + 9 x ( x^{2} + 2 x + 2 ) + 3$	$C_3\wr C_2^2$ (as 12T130)	$324$	$3$	$[\frac{3}{2}, 2, 2, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},1,1,\frac{3}{2}]_{2}^{2}$	$[\frac{3}{2},2,2]$	$[\frac{1}{2},1,1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.17	18	$( x^{2} + 2 x + 2 )^{6} + 9 ( x^{2} + 2 x + 2 )^{3} + 9 x ( x^{2} + 2 x + 2 )^{2} + 9 x ( x^{2} + 2 x + 2 ) + 3$	$C_3\wr C_2^2$ (as 12T130)	$324$	$3$	$[\frac{3}{2}, 2, 2, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},1,1,\frac{3}{2}]_{2}^{2}$	$[\frac{3}{2},2,2]$	$[\frac{1}{2},1,1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.18	18	$( x^{2} + 2 x + 2 )^{6} + 18 ( x^{2} + 2 x + 2 )^{3} + 9 x ( x^{2} + 2 x + 2 )^{2} + 9 x ( x^{2} + 2 x + 2 ) + 3$	$C_3\wr C_2^2$ (as 12T130)	$324$	$3$	$[\frac{3}{2}, 2, 2, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},1,1,\frac{3}{2}]_{2}^{2}$	$[\frac{3}{2},2,2]$	$[\frac{1}{2},1,1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.19	9	$( x^{2} + 2 x + 2 )^{6} + 18 x ( x^{2} + 2 x + 2 )^{2} + 9 x ( x^{2} + 2 x + 2 ) + 3$	$C_3\times S_3^2$ (as 12T70)	$108$	$3$	$[2, 2, \frac{5}{2}]_{2}^{2}$	$[1,1,\frac{3}{2}]_{2}^{2}$	$[2,2]$	$[1,1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.20	9	$( x^{2} + 2 x + 2 )^{6} + 9 ( x^{2} + 2 x + 2 )^{3} + 18 x ( x^{2} + 2 x + 2 )^{2} + 9 x ( x^{2} + 2 x + 2 ) + 3$	$C_3\times S_3^2$ (as 12T70)	$108$	$3$	$[2, 2, \frac{5}{2}]_{2}^{2}$	$[1,1,\frac{3}{2}]_{2}^{2}$	$[2,2]$	$[1,1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.21	9	$( x^{2} + 2 x + 2 )^{6} + 18 ( x^{2} + 2 x + 2 )^{3} + 18 x ( x^{2} + 2 x + 2 )^{2} + 9 x ( x^{2} + 2 x + 2 ) + 3$	$C_3\times S_3^2$ (as 12T70)	$108$	$3$	$[2, 2, \frac{5}{2}]_{2}^{2}$	$[1,1,\frac{3}{2}]_{2}^{2}$	$[2,2]$	$[1,1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.22	18	$( x^{2} + 2 x + 2 )^{6} + 9 ( x^{2} + 2 x + 2 )^{2} + 9 x ( x^{2} + 2 x + 2 ) + 3$	$C_3\wr C_2^2$ (as 12T130)	$324$	$3$	$[\frac{3}{2}, 2, 2, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},1,1,\frac{3}{2}]_{2}^{2}$	$[\frac{3}{2},2,2]$	$[\frac{1}{2},1,1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.23	18	$( x^{2} + 2 x + 2 )^{6} + 9 ( x^{2} + 2 x + 2 )^{3} + 9 ( x^{2} + 2 x + 2 )^{2} + 9 x ( x^{2} + 2 x + 2 ) + 3$	$C_3\wr C_2^2$ (as 12T130)	$324$	$3$	$[\frac{3}{2}, 2, 2, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},1,1,\frac{3}{2}]_{2}^{2}$	$[\frac{3}{2},2,2]$	$[\frac{1}{2},1,1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.24	18	$( x^{2} + 2 x + 2 )^{6} + 18 ( x^{2} + 2 x + 2 )^{3} + 9 ( x^{2} + 2 x + 2 )^{2} + 9 x ( x^{2} + 2 x + 2 ) + 3$	$C_3\wr C_2^2$ (as 12T130)	$324$	$3$	$[\frac{3}{2}, 2, 2, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},1,1,\frac{3}{2}]_{2}^{2}$	$[\frac{3}{2},2,2]$	$[\frac{1}{2},1,1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.25	18	$( x^{2} + 2 x + 2 )^{6} + \left(9 x + 9\right) ( x^{2} + 2 x + 2 )^{2} + 9 x ( x^{2} + 2 x + 2 ) + 3$	$C_3\wr C_2^2$ (as 12T130)	$324$	$3$	$[\frac{3}{2}, 2, 2, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},1,1,\frac{3}{2}]_{2}^{2}$	$[\frac{3}{2},2,2]$	$[\frac{1}{2},1,1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.26	18	$( x^{2} + 2 x + 2 )^{6} + 9 ( x^{2} + 2 x + 2 )^{3} + \left(9 x + 9\right) ( x^{2} + 2 x + 2 )^{2} + 9 x ( x^{2} + 2 x + 2 ) + 3$	$C_3\wr C_2^2$ (as 12T130)	$324$	$3$	$[\frac{3}{2}, 2, 2, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},1,1,\frac{3}{2}]_{2}^{2}$	$[\frac{3}{2},2,2]$	$[\frac{1}{2},1,1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.27	18	$( x^{2} + 2 x + 2 )^{6} + 18 ( x^{2} + 2 x + 2 )^{3} + \left(9 x + 9\right) ( x^{2} + 2 x + 2 )^{2} + 9 x ( x^{2} + 2 x + 2 ) + 3$	$C_3\wr C_2^2$ (as 12T130)	$324$	$3$	$[\frac{3}{2}, 2, 2, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},1,1,\frac{3}{2}]_{2}^{2}$	$[\frac{3}{2},2,2]$	$[\frac{1}{2},1,1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.28	9	$( x^{2} + 2 x + 2 )^{6} + \left(18 x + 9\right) ( x^{2} + 2 x + 2 )^{2} + 9 x ( x^{2} + 2 x + 2 ) + 3$	$C_3\times S_3^2$ (as 12T70)	$108$	$3$	$[2, 2, \frac{5}{2}]_{2}^{2}$	$[1,1,\frac{3}{2}]_{2}^{2}$	$[2,2]$	$[1,1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.29	9	$( x^{2} + 2 x + 2 )^{6} + 9 ( x^{2} + 2 x + 2 )^{3} + \left(18 x + 9\right) ( x^{2} + 2 x + 2 )^{2} + 9 x ( x^{2} + 2 x + 2 ) + 3$	$C_3\times S_3^2$ (as 12T70)	$108$	$3$	$[2, 2, \frac{5}{2}]_{2}^{2}$	$[1,1,\frac{3}{2}]_{2}^{2}$	$[2,2]$	$[1,1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.30	9	$( x^{2} + 2 x + 2 )^{6} + 18 ( x^{2} + 2 x + 2 )^{3} + \left(18 x + 9\right) ( x^{2} + 2 x + 2 )^{2} + 9 x ( x^{2} + 2 x + 2 ) + 3$	$C_3\times S_3^2$ (as 12T70)	$108$	$3$	$[2, 2, \frac{5}{2}]_{2}^{2}$	$[1,1,\frac{3}{2}]_{2}^{2}$	$[2,2]$	$[1,1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.31	18	$( x^{2} + 2 x + 2 )^{6} + 18 ( x^{2} + 2 x + 2 )^{2} + 9 x ( x^{2} + 2 x + 2 ) + 3$	$C_3\wr C_2^2$ (as 12T130)	$324$	$3$	$[\frac{3}{2}, 2, 2, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},1,1,\frac{3}{2}]_{2}^{2}$	$[\frac{3}{2},2,2]$	$[\frac{1}{2},1,1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.32	18	$( x^{2} + 2 x + 2 )^{6} + 9 ( x^{2} + 2 x + 2 )^{3} + 18 ( x^{2} + 2 x + 2 )^{2} + 9 x ( x^{2} + 2 x + 2 ) + 3$	$C_3\wr C_2^2$ (as 12T130)	$324$	$3$	$[\frac{3}{2}, 2, 2, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},1,1,\frac{3}{2}]_{2}^{2}$	$[\frac{3}{2},2,2]$	$[\frac{1}{2},1,1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.33	18	$( x^{2} + 2 x + 2 )^{6} + 18 ( x^{2} + 2 x + 2 )^{3} + 18 ( x^{2} + 2 x + 2 )^{2} + 9 x ( x^{2} + 2 x + 2 ) + 3$	$C_3\wr C_2^2$ (as 12T130)	$324$	$3$	$[\frac{3}{2}, 2, 2, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},1,1,\frac{3}{2}]_{2}^{2}$	$[\frac{3}{2},2,2]$	$[\frac{1}{2},1,1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.34	18	$( x^{2} + 2 x + 2 )^{6} + \left(9 x + 18\right) ( x^{2} + 2 x + 2 )^{2} + 9 x ( x^{2} + 2 x + 2 ) + 3$	$C_3\wr C_2^2$ (as 12T130)	$324$	$3$	$[\frac{3}{2}, 2, 2, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},1,1,\frac{3}{2}]_{2}^{2}$	$[\frac{3}{2},2,2]$	$[\frac{1}{2},1,1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.35	18	$( x^{2} + 2 x + 2 )^{6} + 9 ( x^{2} + 2 x + 2 )^{3} + \left(9 x + 18\right) ( x^{2} + 2 x + 2 )^{2} + 9 x ( x^{2} + 2 x + 2 ) + 3$	$C_3\wr C_2^2$ (as 12T130)	$324$	$3$	$[\frac{3}{2}, 2, 2, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},1,1,\frac{3}{2}]_{2}^{2}$	$[\frac{3}{2},2,2]$	$[\frac{1}{2},1,1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.36	18	$( x^{2} + 2 x + 2 )^{6} + 18 ( x^{2} + 2 x + 2 )^{3} + \left(9 x + 18\right) ( x^{2} + 2 x + 2 )^{2} + 9 x ( x^{2} + 2 x + 2 ) + 3$	$C_3\wr C_2^2$ (as 12T130)	$324$	$3$	$[\frac{3}{2}, 2, 2, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},1,1,\frac{3}{2}]_{2}^{2}$	$[\frac{3}{2},2,2]$	$[\frac{1}{2},1,1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.37	9	$( x^{2} + 2 x + 2 )^{6} + \left(18 x + 18\right) ( x^{2} + 2 x + 2 )^{2} + 9 x ( x^{2} + 2 x + 2 ) + 3$	$C_3\times S_3^2$ (as 12T70)	$108$	$3$	$[2, 2, \frac{5}{2}]_{2}^{2}$	$[1,1,\frac{3}{2}]_{2}^{2}$	$[2,2]$	$[1,1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.38	9	$( x^{2} + 2 x + 2 )^{6} + 9 ( x^{2} + 2 x + 2 )^{3} + \left(18 x + 18\right) ( x^{2} + 2 x + 2 )^{2} + 9 x ( x^{2} + 2 x + 2 ) + 3$	$C_3\times S_3^2$ (as 12T70)	$108$	$3$	$[2, 2, \frac{5}{2}]_{2}^{2}$	$[1,1,\frac{3}{2}]_{2}^{2}$	$[2,2]$	$[1,1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.39	9	$( x^{2} + 2 x + 2 )^{6} + 18 ( x^{2} + 2 x + 2 )^{3} + \left(18 x + 18\right) ( x^{2} + 2 x + 2 )^{2} + 9 x ( x^{2} + 2 x + 2 ) + 3$	$C_3\times S_3^2$ (as 12T70)	$108$	$3$	$[2, 2, \frac{5}{2}]_{2}^{2}$	$[1,1,\frac{3}{2}]_{2}^{2}$	$[2,2]$	$[1,1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.40	9	$( x^{2} + 2 x + 2 )^{6} + 9 ( x^{2} + 2 x + 2 ) + 3$	$C_6\times S_3$ (as 12T18)	$36$	$6$	$[2, \frac{5}{2}]_{2}^{2}$	$[1,\frac{3}{2}]_{2}^{2}$	$[2]$	$[1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.41	9	$( x^{2} + 2 x + 2 )^{6} + 9 ( x^{2} + 2 x + 2 )^{3} + 9 ( x^{2} + 2 x + 2 ) + 3$	$C_6\times S_3$ (as 12T18)	$36$	$6$	$[2, \frac{5}{2}]_{2}^{2}$	$[1,\frac{3}{2}]_{2}^{2}$	$[2]$	$[1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.42	9	$( x^{2} + 2 x + 2 )^{6} + 18 ( x^{2} + 2 x + 2 )^{3} + 9 ( x^{2} + 2 x + 2 ) + 3$	$C_6\times S_3$ (as 12T18)	$36$	$6$	$[2, \frac{5}{2}]_{2}^{2}$	$[1,\frac{3}{2}]_{2}^{2}$	$[2]$	$[1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.43	9	$( x^{2} + 2 x + 2 )^{6} + 9 x ( x^{2} + 2 x + 2 )^{2} + 9 ( x^{2} + 2 x + 2 ) + 3$	$C_3\times S_3^2$ (as 12T70)	$108$	$3$	$[\frac{3}{2}, 2, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},1,\frac{3}{2}]_{2}^{2}$	$[\frac{3}{2},2]$	$[\frac{1}{2},1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.44	9	$( x^{2} + 2 x + 2 )^{6} + 9 ( x^{2} + 2 x + 2 )^{3} + 9 x ( x^{2} + 2 x + 2 )^{2} + 9 ( x^{2} + 2 x + 2 ) + 3$	$C_3\times S_3^2$ (as 12T70)	$108$	$3$	$[\frac{3}{2}, 2, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},1,\frac{3}{2}]_{2}^{2}$	$[\frac{3}{2},2]$	$[\frac{1}{2},1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.45	9	$( x^{2} + 2 x + 2 )^{6} + 18 ( x^{2} + 2 x + 2 )^{3} + 9 x ( x^{2} + 2 x + 2 )^{2} + 9 ( x^{2} + 2 x + 2 ) + 3$	$C_3\times S_3^2$ (as 12T70)	$108$	$3$	$[\frac{3}{2}, 2, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},1,\frac{3}{2}]_{2}^{2}$	$[\frac{3}{2},2]$	$[\frac{1}{2},1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.46	9	$( x^{2} + 2 x + 2 )^{6} + 9 ( x^{2} + 2 x + 2 )^{2} + 9 ( x^{2} + 2 x + 2 ) + 3$	$C_6\times S_3$ (as 12T18)	$36$	$6$	$[2, \frac{5}{2}]_{2}^{2}$	$[1,\frac{3}{2}]_{2}^{2}$	$[2]$	$[1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.47	9	$( x^{2} + 2 x + 2 )^{6} + 9 ( x^{2} + 2 x + 2 )^{3} + 9 ( x^{2} + 2 x + 2 )^{2} + 9 ( x^{2} + 2 x + 2 ) + 3$	$C_6\times S_3$ (as 12T18)	$36$	$6$	$[2, \frac{5}{2}]_{2}^{2}$	$[1,\frac{3}{2}]_{2}^{2}$	$[2]$	$[1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.48	9	$( x^{2} + 2 x + 2 )^{6} + 18 ( x^{2} + 2 x + 2 )^{3} + 9 ( x^{2} + 2 x + 2 )^{2} + 9 ( x^{2} + 2 x + 2 ) + 3$	$C_6\times S_3$ (as 12T18)	$36$	$6$	$[2, \frac{5}{2}]_{2}^{2}$	$[1,\frac{3}{2}]_{2}^{2}$	$[2]$	$[1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.49	9	$( x^{2} + 2 x + 2 )^{6} + \left(9 x + 9\right) ( x^{2} + 2 x + 2 )^{2} + 9 ( x^{2} + 2 x + 2 ) + 3$	$C_3\times S_3^2$ (as 12T70)	$108$	$3$	$[\frac{3}{2}, 2, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},1,\frac{3}{2}]_{2}^{2}$	$[\frac{3}{2},2]$	$[\frac{1}{2},1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$
3.2.6.22a1.50	9	$( x^{2} + 2 x + 2 )^{6} + 9 ( x^{2} + 2 x + 2 )^{3} + \left(9 x + 9\right) ( x^{2} + 2 x + 2 )^{2} + 9 ( x^{2} + 2 x + 2 ) + 3$	$C_3\times S_3^2$ (as 12T70)	$108$	$3$	$[\frac{3}{2}, 2, \frac{5}{2}]_{2}^{2}$	$[\frac{1}{2},1,\frac{3}{2}]_{2}^{2}$	$[\frac{3}{2},2]$	$[\frac{1}{2},1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 2$	$[1, 7]$

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	3.2.6.22a
Base	3.1.1.0a1.1
Degree	\(12\)
e	\(6\)
f	\(2\)
c	\(22\)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Defining polynomial over unramified subextension

Invariants

Diagrams

Varying

Galois groups and Hidden Artin slopes

Fields

Showing 1-50 of 96

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