# Properties

 Label 12.203...752.18t315.a.a Dimension $12$ Group $S_3\wr S_3$ Conductor $2.030\times 10^{17}$ Root number $1$ Indicator $1$

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## Basic invariants

 Dimension: $12$ Group: $S_3\wr S_3$ Conductor: $$203031876107874752$$$$\medspace = 2^{6} \cdot 23^{5} \cdot 149^{4}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 9.1.2668563776.1 Galois orbit size: $1$ Smallest permutation container: 18T315 Parity: odd Determinant: 1.23.2t1.a.a Projective image: $S_3\wr S_3$ Projective stem field: Galois closure of 9.1.2668563776.1

## Defining polynomial

 $f(x)$ $=$ $$x^{9} - x^{8} - x^{7} - 4x^{6} + 7x^{4} + 7x^{3} + 5x^{2} + 4x - 1$$ x^9 - x^8 - x^7 - 4*x^6 + 7*x^4 + 7*x^3 + 5*x^2 + 4*x - 1 .

The roots of $f$ are computed in an extension of $\Q_{ 127 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 127 }$: $$x^{3} + 3x + 124$$

Roots:
 $r_{ 1 }$ $=$ $$5 a^{2} + 49 a + 62 + \left(95 a^{2} + 54 a + 59\right)\cdot 127 + \left(112 a^{2} + 30 a + 76\right)\cdot 127^{2} + \left(117 a^{2} + 79 a + 17\right)\cdot 127^{3} + \left(119 a^{2} + 124 a + 80\right)\cdot 127^{4} + \left(103 a^{2} + 47 a + 15\right)\cdot 127^{5} + \left(58 a^{2} + 96 a + 21\right)\cdot 127^{6} + \left(a^{2} + 76 a + 121\right)\cdot 127^{7} + \left(27 a^{2} + 126 a + 45\right)\cdot 127^{8} + \left(74 a^{2} + 126 a + 18\right)\cdot 127^{9} +O(127^{10})$$ 5*a^2 + 49*a + 62 + (95*a^2 + 54*a + 59)*127 + (112*a^2 + 30*a + 76)*127^2 + (117*a^2 + 79*a + 17)*127^3 + (119*a^2 + 124*a + 80)*127^4 + (103*a^2 + 47*a + 15)*127^5 + (58*a^2 + 96*a + 21)*127^6 + (a^2 + 76*a + 121)*127^7 + (27*a^2 + 126*a + 45)*127^8 + (74*a^2 + 126*a + 18)*127^9+O(127^10) $r_{ 2 }$ $=$ $$17 a^{2} + 90 a + 75 + \left(45 a^{2} + 88 a + 95\right)\cdot 127 + \left(77 a^{2} + 99 a + 48\right)\cdot 127^{2} + \left(17 a^{2} + 11 a + 107\right)\cdot 127^{3} + \left(62 a^{2} + 95 a + 118\right)\cdot 127^{4} + \left(69 a^{2} + 92 a + 36\right)\cdot 127^{5} + \left(27 a^{2} + 97 a + 28\right)\cdot 127^{6} + \left(14 a^{2} + 126 a + 4\right)\cdot 127^{7} + \left(4 a^{2} + 84 a + 57\right)\cdot 127^{8} + \left(91 a^{2} + 68 a + 118\right)\cdot 127^{9} +O(127^{10})$$ 17*a^2 + 90*a + 75 + (45*a^2 + 88*a + 95)*127 + (77*a^2 + 99*a + 48)*127^2 + (17*a^2 + 11*a + 107)*127^3 + (62*a^2 + 95*a + 118)*127^4 + (69*a^2 + 92*a + 36)*127^5 + (27*a^2 + 97*a + 28)*127^6 + (14*a^2 + 126*a + 4)*127^7 + (4*a^2 + 84*a + 57)*127^8 + (91*a^2 + 68*a + 118)*127^9+O(127^10) $r_{ 3 }$ $=$ $$19 a^{2} + 33 a + 90 + \left(54 a^{2} + 32 a + 104\right)\cdot 127 + \left(8 a^{2} + 98 a + 121\right)\cdot 127^{2} + \left(109 a^{2} + 93 a + 126\right)\cdot 127^{3} + \left(29 a^{2} + 55 a + 26\right)\cdot 127^{4} + \left(81 a^{2} + 58 a + 97\right)\cdot 127^{5} + \left(46 a^{2} + 11 a + 123\right)\cdot 127^{6} + \left(114 a^{2} + 39 a + 92\right)\cdot 127^{7} + \left(57 a^{2} + 38 a + 107\right)\cdot 127^{8} + \left(23 a^{2} + 23 a + 43\right)\cdot 127^{9} +O(127^{10})$$ 19*a^2 + 33*a + 90 + (54*a^2 + 32*a + 104)*127 + (8*a^2 + 98*a + 121)*127^2 + (109*a^2 + 93*a + 126)*127^3 + (29*a^2 + 55*a + 26)*127^4 + (81*a^2 + 58*a + 97)*127^5 + (46*a^2 + 11*a + 123)*127^6 + (114*a^2 + 39*a + 92)*127^7 + (57*a^2 + 38*a + 107)*127^8 + (23*a^2 + 23*a + 43)*127^9+O(127^10) $r_{ 4 }$ $=$ $$30 a^{2} + 92 a + 101 + \left(14 a^{2} + 49 a + 33\right)\cdot 127 + \left(121 a^{2} + 6 a + 9\right)\cdot 127^{2} + \left(72 a^{2} + 125 a + 91\right)\cdot 127^{3} + \left(44 a^{2} + 32 a + 83\right)\cdot 127^{4} + \left(105 a^{2} + 105 a + 108\right)\cdot 127^{5} + \left(71 a^{2} + 82 a + 116\right)\cdot 127^{6} + \left(104 a^{2} + 117 a + 57\right)\cdot 127^{7} + \left(81 a^{2} + 101 a + 85\right)\cdot 127^{8} + \left(42 a^{2} + 108 a + 21\right)\cdot 127^{9} +O(127^{10})$$ 30*a^2 + 92*a + 101 + (14*a^2 + 49*a + 33)*127 + (121*a^2 + 6*a + 9)*127^2 + (72*a^2 + 125*a + 91)*127^3 + (44*a^2 + 32*a + 83)*127^4 + (105*a^2 + 105*a + 108)*127^5 + (71*a^2 + 82*a + 116)*127^6 + (104*a^2 + 117*a + 57)*127^7 + (81*a^2 + 101*a + 85)*127^8 + (42*a^2 + 108*a + 21)*127^9+O(127^10) $r_{ 5 }$ $=$ $$57 a^{2} + 120 a + 106 + \left(51 a^{2} + 96 a + 58\right)\cdot 127 + \left(48 a^{2} + 67 a + 55\right)\cdot 127^{2} + \left(62 a^{2} + 120 a + 101\right)\cdot 127^{3} + \left(81 a^{2} + 67 a + 31\right)\cdot 127^{4} + \left(91 a^{2} + 91 a + 54\right)\cdot 127^{5} + \left(78 a^{2} + 73 a + 111\right)\cdot 127^{6} + \left(92 a^{2} + 42 a + 48\right)\cdot 127^{7} + \left(88 a^{2} + 46 a + 94\right)\cdot 127^{8} + \left(118 a^{2} + 12 a + 7\right)\cdot 127^{9} +O(127^{10})$$ 57*a^2 + 120*a + 106 + (51*a^2 + 96*a + 58)*127 + (48*a^2 + 67*a + 55)*127^2 + (62*a^2 + 120*a + 101)*127^3 + (81*a^2 + 67*a + 31)*127^4 + (91*a^2 + 91*a + 54)*127^5 + (78*a^2 + 73*a + 111)*127^6 + (92*a^2 + 42*a + 48)*127^7 + (88*a^2 + 46*a + 94)*127^8 + (118*a^2 + 12*a + 7)*127^9+O(127^10) $r_{ 6 }$ $=$ $$72 a^{2} + 23 a + 9 + \left(19 a^{2} + 93 a + 122\right)\cdot 127 + \left(a^{2} + 111 a + 87\right)\cdot 127^{2} + \left(29 a^{2} + 71 a + 34\right)\cdot 127^{3} + \left(61 a^{2} + 118\right)\cdot 127^{4} + \left(89 a^{2} + 32 a + 49\right)\cdot 127^{5} + \left(40 a^{2} + 81 a + 35\right)\cdot 127^{6} + \left(109 a^{2} + 81 a + 82\right)\cdot 127^{7} + \left(86 a^{2} + 57 a + 90\right)\cdot 127^{8} + \left(77 a^{2} + a + 52\right)\cdot 127^{9} +O(127^{10})$$ 72*a^2 + 23*a + 9 + (19*a^2 + 93*a + 122)*127 + (a^2 + 111*a + 87)*127^2 + (29*a^2 + 71*a + 34)*127^3 + (61*a^2 + 118)*127^4 + (89*a^2 + 32*a + 49)*127^5 + (40*a^2 + 81*a + 35)*127^6 + (109*a^2 + 81*a + 82)*127^7 + (86*a^2 + 57*a + 90)*127^8 + (77*a^2 + a + 52)*127^9+O(127^10) $r_{ 7 }$ $=$ $$80 a^{2} + 72 a + 74 + \left(67 a^{2} + 115 a + 13\right)\cdot 127 + \left(55 a^{2} + 20 a + 5\right)\cdot 127^{2} + \left(36 a^{2} + 117 a + 18\right)\cdot 127^{3} + \left(20 a^{2} + 125 a + 35\right)\cdot 127^{4} + \left(79 a^{2} + 55 a + 56\right)\cdot 127^{5} + \left(27 a^{2} + 73 a + 28\right)\cdot 127^{6} + \left(8 a^{2} + 9 a + 119\right)\cdot 127^{7} + \left(41 a^{2} + 67 a + 3\right)\cdot 127^{8} + \left(120 a^{2} + 76 a + 50\right)\cdot 127^{9} +O(127^{10})$$ 80*a^2 + 72*a + 74 + (67*a^2 + 115*a + 13)*127 + (55*a^2 + 20*a + 5)*127^2 + (36*a^2 + 117*a + 18)*127^3 + (20*a^2 + 125*a + 35)*127^4 + (79*a^2 + 55*a + 56)*127^5 + (27*a^2 + 73*a + 28)*127^6 + (8*a^2 + 9*a + 119)*127^7 + (41*a^2 + 67*a + 3)*127^8 + (120*a^2 + 76*a + 50)*127^9+O(127^10) $r_{ 8 }$ $=$ $$103 a^{2} + 45 a + 4 + \left(104 a^{2} + 40 a + 79\right)\cdot 127 + \left(5 a^{2} + 125 a + 116\right)\cdot 127^{2} + \left(27 a^{2} + 80 a + 89\right)\cdot 127^{3} + \left(104 a^{2} + 73 a + 48\right)\cdot 127^{4} + \left(68 a^{2} + 20 a + 72\right)\cdot 127^{5} + \left(21 a^{2} + 19 a + 73\right)\cdot 127^{6} + \left(11 a^{2} + 11 a + 13\right)\cdot 127^{7} + \left(42 a^{2} + 89 a + 76\right)\cdot 127^{8} + \left(29 a^{2} + 103 a + 55\right)\cdot 127^{9} +O(127^{10})$$ 103*a^2 + 45*a + 4 + (104*a^2 + 40*a + 79)*127 + (5*a^2 + 125*a + 116)*127^2 + (27*a^2 + 80*a + 89)*127^3 + (104*a^2 + 73*a + 48)*127^4 + (68*a^2 + 20*a + 72)*127^5 + (21*a^2 + 19*a + 73)*127^6 + (11*a^2 + 11*a + 13)*127^7 + (42*a^2 + 89*a + 76)*127^8 + (29*a^2 + 103*a + 55)*127^9+O(127^10) $r_{ 9 }$ $=$ $$125 a^{2} + 111 a + 115 + \left(55 a^{2} + 63 a + 67\right)\cdot 127 + \left(77 a^{2} + 74 a + 113\right)\cdot 127^{2} + \left(35 a^{2} + 61 a + 47\right)\cdot 127^{3} + \left(111 a^{2} + 58 a + 91\right)\cdot 127^{4} + \left(72 a^{2} + 3 a + 16\right)\cdot 127^{5} + \left(7 a^{2} + 99 a + 96\right)\cdot 127^{6} + \left(52 a^{2} + 2 a + 94\right)\cdot 127^{7} + \left(78 a^{2} + 23 a + 73\right)\cdot 127^{8} + \left(57 a^{2} + 113 a + 12\right)\cdot 127^{9} +O(127^{10})$$ 125*a^2 + 111*a + 115 + (55*a^2 + 63*a + 67)*127 + (77*a^2 + 74*a + 113)*127^2 + (35*a^2 + 61*a + 47)*127^3 + (111*a^2 + 58*a + 91)*127^4 + (72*a^2 + 3*a + 16)*127^5 + (7*a^2 + 99*a + 96)*127^6 + (52*a^2 + 2*a + 94)*127^7 + (78*a^2 + 23*a + 73)*127^8 + (57*a^2 + 113*a + 12)*127^9+O(127^10)

## Generators of the action on the roots $r_1, \ldots, r_{ 9 }$

 Cycle notation $(5,7)$ $(1,5,2)(3,4,7)(6,8,9)$ $(1,4,6)$ $(2,5)(3,7)(8,9)$ $(2,3,9)$ $(5,7,8)$

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 9 }$ Character value $1$ $1$ $()$ $12$ $9$ $2$ $(5,7)$ $4$ $18$ $2$ $(2,5)(3,7)(8,9)$ $2$ $27$ $2$ $(1,4)(2,3)(5,7)$ $0$ $27$ $2$ $(1,4)(5,7)$ $0$ $54$ $2$ $(1,5)(2,3)(4,7)(6,8)$ $2$ $6$ $3$ $(1,4,6)$ $0$ $8$ $3$ $(1,6,4)(2,9,3)(5,8,7)$ $3$ $12$ $3$ $(1,6,4)(5,8,7)$ $-3$ $72$ $3$ $(1,5,2)(3,4,7)(6,8,9)$ $0$ $54$ $4$ $(1,7,4,5)(6,8)$ $0$ $162$ $4$ $(1,7,4,5)(2,3)(6,8)$ $0$ $36$ $6$ $(1,4,6)(2,5)(3,7)(8,9)$ $2$ $36$ $6$ $(1,8,6,7,4,5)$ $-1$ $36$ $6$ $(1,4,6)(5,7)$ $-2$ $36$ $6$ $(1,4,6)(2,3,9)(5,7)$ $1$ $54$ $6$ $(1,6,4)(2,3)(5,7)$ $0$ $72$ $6$ $(1,4,6)(2,8,9,7,3,5)$ $-1$ $108$ $6$ $(1,8,6,7,4,5)(2,3)$ $-1$ $216$ $6$ $(1,7,3,4,5,2)(6,8,9)$ $0$ $144$ $9$ $(1,8,9,6,7,3,4,5,2)$ $0$ $108$ $12$ $(1,4,6)(2,7,3,5)(8,9)$ $0$

The blue line marks the conjugacy class containing complex conjugation.