# Properties

 Label 12.539...152.18t315.a Dimension $12$ Group $S_3\wr S_3$ Conductor $5.400\times 10^{16}$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $12$ Group: $S_3\wr S_3$ Conductor: $$53995089429705152$$$$\medspace = 2^{6} \cdot 23^{5} \cdot 107^{4}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 9.1.83319616.1 Galois orbit size: $1$ Smallest permutation container: 18T315 Parity: odd Projective image: $S_3\wr S_3$ Projective field: Galois closure of 9.1.83319616.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$: $$x^{3} + 3x + 99$$
Roots:
 $r_{ 1 }$ $=$ $$45 a^{2} + 93 a + 92 + \left(99 a^{2} + 54 a + 25\right)\cdot 101 + \left(2 a^{2} + 78 a + 2\right)\cdot 101^{2} + \left(3 a^{2} + 44 a + 2\right)\cdot 101^{3} + \left(64 a^{2} + 96 a + 47\right)\cdot 101^{4} + \left(66 a^{2} + 56 a + 2\right)\cdot 101^{5} + \left(67 a^{2} + 13 a + 62\right)\cdot 101^{6} + \left(63 a^{2} + 50 a + 60\right)\cdot 101^{7} + \left(a^{2} + 82 a + 59\right)\cdot 101^{8} + \left(63 a^{2} + 33 a + 71\right)\cdot 101^{9} +O(101^{10})$$ 45*a^2 + 93*a + 92 + (99*a^2 + 54*a + 25)*101 + (2*a^2 + 78*a + 2)*101^2 + (3*a^2 + 44*a + 2)*101^3 + (64*a^2 + 96*a + 47)*101^4 + (66*a^2 + 56*a + 2)*101^5 + (67*a^2 + 13*a + 62)*101^6 + (63*a^2 + 50*a + 60)*101^7 + (a^2 + 82*a + 59)*101^8 + (63*a^2 + 33*a + 71)*101^9+O(101^10) $r_{ 2 }$ $=$ $$42 a^{2} + 68 a + 97 + \left(8 a^{2} + 92 a + 1\right)\cdot 101 + \left(74 a^{2} + 47 a + 93\right)\cdot 101^{2} + \left(36 a^{2} + 29 a + 13\right)\cdot 101^{3} + \left(72 a^{2} + 94 a + 38\right)\cdot 101^{4} + \left(88 a^{2} + 16 a + 39\right)\cdot 101^{5} + \left(38 a^{2} + 14 a + 71\right)\cdot 101^{6} + \left(86 a^{2} + 69 a + 54\right)\cdot 101^{7} + \left(90 a^{2} + 83 a + 27\right)\cdot 101^{8} + \left(25 a^{2} + 47 a + 78\right)\cdot 101^{9} +O(101^{10})$$ 42*a^2 + 68*a + 97 + (8*a^2 + 92*a + 1)*101 + (74*a^2 + 47*a + 93)*101^2 + (36*a^2 + 29*a + 13)*101^3 + (72*a^2 + 94*a + 38)*101^4 + (88*a^2 + 16*a + 39)*101^5 + (38*a^2 + 14*a + 71)*101^6 + (86*a^2 + 69*a + 54)*101^7 + (90*a^2 + 83*a + 27)*101^8 + (25*a^2 + 47*a + 78)*101^9+O(101^10) $r_{ 3 }$ $=$ $$90 a^{2} + 86 a + 92 + \left(a^{2} + 71 a + 89\right)\cdot 101 + \left(20 a^{2} + 22 a + 85\right)\cdot 101^{2} + \left(36 a^{2} + 34 a + 12\right)\cdot 101^{3} + \left(58 a^{2} + 50 a + 10\right)\cdot 101^{4} + \left(26 a^{2} + 19 a + 16\right)\cdot 101^{5} + \left(37 a^{2} + 96 a + 68\right)\cdot 101^{6} + \left(88 a^{2} + 45 a + 58\right)\cdot 101^{7} + \left(3 a^{2} + 18 a + 55\right)\cdot 101^{8} + \left(99 a^{2} + 40 a + 22\right)\cdot 101^{9} +O(101^{10})$$ 90*a^2 + 86*a + 92 + (a^2 + 71*a + 89)*101 + (20*a^2 + 22*a + 85)*101^2 + (36*a^2 + 34*a + 12)*101^3 + (58*a^2 + 50*a + 10)*101^4 + (26*a^2 + 19*a + 16)*101^5 + (37*a^2 + 96*a + 68)*101^6 + (88*a^2 + 45*a + 58)*101^7 + (3*a^2 + 18*a + 55)*101^8 + (99*a^2 + 40*a + 22)*101^9+O(101^10) $r_{ 4 }$ $=$ $$57 + 52\cdot 101 + 18\cdot 101^{2} + 15\cdot 101^{3} + 43\cdot 101^{4} + 46\cdot 101^{5} + 37\cdot 101^{6} + 36\cdot 101^{7} + 9\cdot 101^{8} + 16\cdot 101^{9} +O(101^{10})$$ 57 + 52*101 + 18*101^2 + 15*101^3 + 43*101^4 + 46*101^5 + 37*101^6 + 36*101^7 + 9*101^8 + 16*101^9+O(101^10) $r_{ 5 }$ $=$ $$42 + 90\cdot 101 + 99\cdot 101^{2} + 84\cdot 101^{3} + 86\cdot 101^{4} + 63\cdot 101^{5} + 53\cdot 101^{6} + 60\cdot 101^{7} + 69\cdot 101^{8} + 81\cdot 101^{9} +O(101^{10})$$ 42 + 90*101 + 99*101^2 + 84*101^3 + 86*101^4 + 63*101^5 + 53*101^6 + 60*101^7 + 69*101^8 + 81*101^9+O(101^10) $r_{ 6 }$ $=$ $$70 a^{2} + 48 a + 52 + \left(90 a^{2} + 37 a + 65\right)\cdot 101 + \left(6 a^{2} + 30 a + 59\right)\cdot 101^{2} + \left(28 a^{2} + 37 a + 97\right)\cdot 101^{3} + \left(71 a^{2} + 57 a + 35\right)\cdot 101^{4} + \left(86 a^{2} + 64 a + 35\right)\cdot 101^{5} + \left(24 a^{2} + 91 a + 43\right)\cdot 101^{6} + \left(27 a^{2} + 86 a + 37\right)\cdot 101^{7} + \left(6 a^{2} + 99 a + 60\right)\cdot 101^{8} + \left(77 a^{2} + 12 a + 79\right)\cdot 101^{9} +O(101^{10})$$ 70*a^2 + 48*a + 52 + (90*a^2 + 37*a + 65)*101 + (6*a^2 + 30*a + 59)*101^2 + (28*a^2 + 37*a + 97)*101^3 + (71*a^2 + 57*a + 35)*101^4 + (86*a^2 + 64*a + 35)*101^5 + (24*a^2 + 91*a + 43)*101^6 + (27*a^2 + 86*a + 37)*101^7 + (6*a^2 + 99*a + 60)*101^8 + (77*a^2 + 12*a + 79)*101^9+O(101^10) $r_{ 7 }$ $=$ $$62 + 16\cdot 101 + 58\cdot 101^{2} + 90\cdot 101^{3} + 28\cdot 101^{4} + 90\cdot 101^{5} + 46\cdot 101^{6} + 53\cdot 101^{7} + 12\cdot 101^{8} + 87\cdot 101^{9} +O(101^{10})$$ 62 + 16*101 + 58*101^2 + 90*101^3 + 28*101^4 + 90*101^5 + 46*101^6 + 53*101^7 + 12*101^8 + 87*101^9+O(101^10) $r_{ 8 }$ $=$ $$62 a^{2} + 26 a + 25 + \left(83 a^{2} + 50 a + 95\right)\cdot 101 + \left(96 a^{2} + 16 a + 88\right)\cdot 101^{2} + \left(50 a^{2} + 80 a + 97\right)\cdot 101^{3} + \left(91 a^{2} + 97 a\right)\cdot 101^{4} + \left(63 a^{2} + 87 a + 98\right)\cdot 101^{5} + \left(25 a^{2} + 75 a + 78\right)\cdot 101^{6} + \left(94 a^{2} + 9 a + 20\right)\cdot 101^{7} + \left(70 a^{2} + 15 a + 97\right)\cdot 101^{8} + \left(40 a^{2} + 8 a + 26\right)\cdot 101^{9} +O(101^{10})$$ 62*a^2 + 26*a + 25 + (83*a^2 + 50*a + 95)*101 + (96*a^2 + 16*a + 88)*101^2 + (50*a^2 + 80*a + 97)*101^3 + (91*a^2 + 97*a)*101^4 + (63*a^2 + 87*a + 98)*101^5 + (25*a^2 + 75*a + 78)*101^6 + (94*a^2 + 9*a + 20)*101^7 + (70*a^2 + 15*a + 97)*101^8 + (40*a^2 + 8*a + 26)*101^9+O(101^10) $r_{ 9 }$ $=$ $$95 a^{2} + 83 a + 91 + \left(18 a^{2} + 96 a + 66\right)\cdot 101 + \left(a^{2} + 5 a + 99\right)\cdot 101^{2} + \left(47 a^{2} + 77 a + 89\right)\cdot 101^{3} + \left(46 a^{2} + 7 a + 11\right)\cdot 101^{4} + \left(71 a^{2} + 57 a + 12\right)\cdot 101^{5} + \left(7 a^{2} + 11 a + 43\right)\cdot 101^{6} + \left(44 a^{2} + 41 a + 21\right)\cdot 101^{7} + \left(28 a^{2} + 3 a + 12\right)\cdot 101^{8} + \left(98 a^{2} + 59 a + 41\right)\cdot 101^{9} +O(101^{10})$$ 95*a^2 + 83*a + 91 + (18*a^2 + 96*a + 66)*101 + (a^2 + 5*a + 99)*101^2 + (47*a^2 + 77*a + 89)*101^3 + (46*a^2 + 7*a + 11)*101^4 + (71*a^2 + 57*a + 12)*101^5 + (7*a^2 + 11*a + 43)*101^6 + (44*a^2 + 41*a + 21)*101^7 + (28*a^2 + 3*a + 12)*101^8 + (98*a^2 + 59*a + 41)*101^9+O(101^10)

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 9 }$ Character values $c1$ $1$ $1$ $()$ $12$ $9$ $2$ $(4,5)$ $4$ $18$ $2$ $(1,4)(5,8)(7,9)$ $2$ $27$ $2$ $(1,8)(2,3)(4,5)$ $0$ $27$ $2$ $(1,8)(4,5)$ $0$ $54$ $2$ $(1,2)(3,8)(4,5)(6,9)$ $2$ $6$ $3$ $(2,3,6)$ $0$ $8$ $3$ $(1,8,9)(2,3,6)(4,5,7)$ $3$ $12$ $3$ $(2,3,6)(4,5,7)$ $-3$ $72$ $3$ $(1,2,4)(3,5,8)(6,7,9)$ $0$ $54$ $4$ $(1,4,8,5)(7,9)$ $0$ $162$ $4$ $(2,4,3,5)(6,7)(8,9)$ $0$ $36$ $6$ $(1,4)(2,3,6)(5,8)(7,9)$ $2$ $36$ $6$ $(2,5,3,7,6,4)$ $-1$ $36$ $6$ $(2,3,6)(4,5)$ $-2$ $36$ $6$ $(1,8,9)(2,3,6)(4,5)$ $1$ $54$ $6$ $(1,8)(2,6,3)(4,5)$ $0$ $72$ $6$ $(1,5,8,7,9,4)(2,3,6)$ $-1$ $108$ $6$ $(1,2,8,3,9,6)(4,5)$ $-1$ $216$ $6$ $(1,2,4,8,3,5)(6,7,9)$ $0$ $144$ $9$ $(1,2,5,8,3,7,9,6,4)$ $0$ $108$ $12$ $(1,4,8,5)(2,3,6)(7,9)$ $0$
The blue line marks the conjugacy class containing complex conjugation.