# Properties

 Label 27.334...376.28t165.a.a Dimension $27$ Group $\mathrm{P}\Gamma\mathrm{L}(2,8)$ Conductor $3.350\times 10^{44}$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $27$ Group: $\mathrm{P}\Gamma\mathrm{L}(2,8)$ Conductor: $$334\!\cdots\!376$$$$\medspace = 2^{30} \cdot 7^{42}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 9.1.72313663744.1 Galois orbit size: $1$ Smallest permutation container: 28T165 Parity: even Determinant: 1.1.1t1.a.a Projective image: $\PSL(2,8).C_3$ Projective stem field: Galois closure of 9.1.72313663744.1

## Defining polynomial

 $f(x)$ $=$ $$x^{9} - x^{8} - 4x^{7} + 28x^{3} + 26x^{2} + 9x + 1$$ x^9 - x^8 - 4*x^7 + 28*x^3 + 26*x^2 + 9*x + 1 .

The roots of $f$ are computed in an extension of $\Q_{ 103 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 103 }$: $$x^{3} + 2x + 98$$

Roots:
 $r_{ 1 }$ $=$ $$34 + 65\cdot 103 + 25\cdot 103^{2} + 85\cdot 103^{3} + 33\cdot 103^{4} + 84\cdot 103^{5} + 15\cdot 103^{6} + 87\cdot 103^{7} + 61\cdot 103^{8} + 33\cdot 103^{9} +O(103^{10})$$ 34 + 65*103 + 25*103^2 + 85*103^3 + 33*103^4 + 84*103^5 + 15*103^6 + 87*103^7 + 61*103^8 + 33*103^9+O(103^10) $r_{ 2 }$ $=$ $$59 + 67\cdot 103 + 95\cdot 103^{2} + 85\cdot 103^{3} + 47\cdot 103^{4} + 64\cdot 103^{5} + 6\cdot 103^{6} + 6\cdot 103^{7} + 43\cdot 103^{8} + 25\cdot 103^{9} +O(103^{10})$$ 59 + 67*103 + 95*103^2 + 85*103^3 + 47*103^4 + 64*103^5 + 6*103^6 + 6*103^7 + 43*103^8 + 25*103^9+O(103^10) $r_{ 3 }$ $=$ $$75 + 23\cdot 103 + 67\cdot 103^{2} + 48\cdot 103^{3} + 68\cdot 103^{4} + 12\cdot 103^{5} + 14\cdot 103^{6} + 21\cdot 103^{7} + 17\cdot 103^{8} + 52\cdot 103^{9} +O(103^{10})$$ 75 + 23*103 + 67*103^2 + 48*103^3 + 68*103^4 + 12*103^5 + 14*103^6 + 21*103^7 + 17*103^8 + 52*103^9+O(103^10) $r_{ 4 }$ $=$ $$32 a + 51 + \left(23 a^{2} + 30 a + 61\right)\cdot 103 + \left(30 a^{2} + 85 a + 43\right)\cdot 103^{2} + \left(13 a^{2} + 42 a + 80\right)\cdot 103^{3} + \left(3 a^{2} + 37 a + 27\right)\cdot 103^{4} + \left(82 a^{2} + 27 a + 2\right)\cdot 103^{5} + \left(96 a^{2} + 52 a + 65\right)\cdot 103^{6} + \left(16 a^{2} + 53 a + 45\right)\cdot 103^{7} + \left(17 a^{2} + 44 a + 50\right)\cdot 103^{8} + \left(67 a^{2} + 18 a + 15\right)\cdot 103^{9} +O(103^{10})$$ 32*a + 51 + (23*a^2 + 30*a + 61)*103 + (30*a^2 + 85*a + 43)*103^2 + (13*a^2 + 42*a + 80)*103^3 + (3*a^2 + 37*a + 27)*103^4 + (82*a^2 + 27*a + 2)*103^5 + (96*a^2 + 52*a + 65)*103^6 + (16*a^2 + 53*a + 45)*103^7 + (17*a^2 + 44*a + 50)*103^8 + (67*a^2 + 18*a + 15)*103^9+O(103^10) $r_{ 5 }$ $=$ $$23 a^{2} + 96 a + 13 + \left(51 a^{2} + 62 a + 99\right)\cdot 103 + \left(72 a^{2} + 58 a + 99\right)\cdot 103^{2} + \left(60 a^{2} + 58 a + 74\right)\cdot 103^{3} + \left(101 a^{2} + 60 a + 21\right)\cdot 103^{4} + \left(72 a^{2} + 73 a + 93\right)\cdot 103^{5} + \left(39 a^{2} + 50 a + 91\right)\cdot 103^{6} + \left(100 a^{2} + 101 a + 53\right)\cdot 103^{7} + \left(31 a^{2} + 17 a + 1\right)\cdot 103^{8} + \left(87 a^{2} + 48 a + 8\right)\cdot 103^{9} +O(103^{10})$$ 23*a^2 + 96*a + 13 + (51*a^2 + 62*a + 99)*103 + (72*a^2 + 58*a + 99)*103^2 + (60*a^2 + 58*a + 74)*103^3 + (101*a^2 + 60*a + 21)*103^4 + (72*a^2 + 73*a + 93)*103^5 + (39*a^2 + 50*a + 91)*103^6 + (100*a^2 + 101*a + 53)*103^7 + (31*a^2 + 17*a + 1)*103^8 + (87*a^2 + 48*a + 8)*103^9+O(103^10) $r_{ 6 }$ $=$ $$24 a^{2} + 51 a + 97 + \left(47 a^{2} + 45 a + 82\right)\cdot 103 + \left(95 a^{2} + 35 a + 26\right)\cdot 103^{2} + \left(81 a + 71\right)\cdot 103^{3} + \left(9 a^{2} + 77 a + 75\right)\cdot 103^{4} + \left(5 a^{2} + 87 a + 25\right)\cdot 103^{5} + \left(66 a^{2} + 76 a + 71\right)\cdot 103^{6} + \left(13 a^{2} + 35 a + 25\right)\cdot 103^{7} + \left(50 a^{2} + 50 a + 67\right)\cdot 103^{8} + \left(66 a^{2} + 51 a + 22\right)\cdot 103^{9} +O(103^{10})$$ 24*a^2 + 51*a + 97 + (47*a^2 + 45*a + 82)*103 + (95*a^2 + 35*a + 26)*103^2 + (81*a + 71)*103^3 + (9*a^2 + 77*a + 75)*103^4 + (5*a^2 + 87*a + 25)*103^5 + (66*a^2 + 76*a + 71)*103^6 + (13*a^2 + 35*a + 25)*103^7 + (50*a^2 + 50*a + 67)*103^8 + (66*a^2 + 51*a + 22)*103^9+O(103^10) $r_{ 7 }$ $=$ $$34 a^{2} + 10 a + 76 + \left(93 a^{2} + 86 a + 75\right)\cdot 103 + \left(92 a^{2} + 13 a + 57\right)\cdot 103^{2} + \left(41 a^{2} + 83 a + 91\right)\cdot 103^{3} + \left(39 a^{2} + 79 a + 81\right)\cdot 103^{4} + \left(45 a^{2} + 87 a + 10\right)\cdot 103^{5} + \left(31 a^{2} + 29 a + 25\right)\cdot 103^{6} + \left(94 a^{2} + 44 a + 30\right)\cdot 103^{7} + \left(91 a^{2} + 76 a + 54\right)\cdot 103^{8} + \left(95 a^{2} + 100 a + 27\right)\cdot 103^{9} +O(103^{10})$$ 34*a^2 + 10*a + 76 + (93*a^2 + 86*a + 75)*103 + (92*a^2 + 13*a + 57)*103^2 + (41*a^2 + 83*a + 91)*103^3 + (39*a^2 + 79*a + 81)*103^4 + (45*a^2 + 87*a + 10)*103^5 + (31*a^2 + 29*a + 25)*103^6 + (94*a^2 + 44*a + 30)*103^7 + (91*a^2 + 76*a + 54)*103^8 + (95*a^2 + 100*a + 27)*103^9+O(103^10) $r_{ 8 }$ $=$ $$45 a^{2} + 42 a + 22 + \left(65 a^{2} + 74 a + 4\right)\cdot 103 + \left(17 a^{2} + 53 a + 26\right)\cdot 103^{2} + \left(60 a^{2} + 41 a + 47\right)\cdot 103^{3} + \left(54 a^{2} + 48 a + 33\right)\cdot 103^{4} + \left(52 a^{2} + 30 a + 20\right)\cdot 103^{5} + \left(5 a^{2} + 99 a + 59\right)\cdot 103^{6} + \left(98 a^{2} + 22 a + 69\right)\cdot 103^{7} + \left(63 a^{2} + 79 a + 85\right)\cdot 103^{8} + \left(43 a^{2} + 53 a + 60\right)\cdot 103^{9} +O(103^{10})$$ 45*a^2 + 42*a + 22 + (65*a^2 + 74*a + 4)*103 + (17*a^2 + 53*a + 26)*103^2 + (60*a^2 + 41*a + 47)*103^3 + (54*a^2 + 48*a + 33)*103^4 + (52*a^2 + 30*a + 20)*103^5 + (5*a^2 + 99*a + 59)*103^6 + (98*a^2 + 22*a + 69)*103^7 + (63*a^2 + 79*a + 85)*103^8 + (43*a^2 + 53*a + 60)*103^9+O(103^10) $r_{ 9 }$ $=$ $$80 a^{2} + 78 a + 89 + \left(28 a^{2} + 9 a + 34\right)\cdot 103 + \left(62 a + 72\right)\cdot 103^{2} + \left(29 a^{2} + a + 32\right)\cdot 103^{3} + \left(101 a^{2} + 5 a + 21\right)\cdot 103^{4} + \left(50 a^{2} + 2 a + 98\right)\cdot 103^{5} + \left(69 a^{2} + 62\right)\cdot 103^{6} + \left(88 a^{2} + 51 a + 72\right)\cdot 103^{7} + \left(53 a^{2} + 40 a + 30\right)\cdot 103^{8} + \left(51 a^{2} + 36 a + 63\right)\cdot 103^{9} +O(103^{10})$$ 80*a^2 + 78*a + 89 + (28*a^2 + 9*a + 34)*103 + (62*a + 72)*103^2 + (29*a^2 + a + 32)*103^3 + (101*a^2 + 5*a + 21)*103^4 + (50*a^2 + 2*a + 98)*103^5 + (69*a^2 + 62)*103^6 + (88*a^2 + 51*a + 72)*103^7 + (53*a^2 + 40*a + 30)*103^8 + (51*a^2 + 36*a + 63)*103^9+O(103^10)

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 9 }$ Character value $1$ $1$ $()$ $27$ $63$ $2$ $(1,3)(2,8)(4,7)(5,9)$ $3$ $56$ $3$ $(1,2,8)(3,6,9)(4,5,7)$ $0$ $84$ $3$ $(2,6,3)(4,5,8)$ $0$ $84$ $3$ $(2,3,6)(4,8,5)$ $0$ $252$ $6$ $(1,2,9,3,8,5)(4,7)$ $0$ $252$ $6$ $(1,5,8,3,9,2)(4,7)$ $0$ $216$ $7$ $(1,3,4,8,6,2,7)$ $-1$ $168$ $9$ $(1,3,7,4,5,8,2,6,9)$ $0$ $168$ $9$ $(1,7,5,2,9,3,4,8,6)$ $0$ $168$ $9$ $(1,3,8,6,9,7,4,2,5)$ $0$

The blue line marks the conjugacy class containing complex conjugation.