# Properties

 Label 8.253...401.12t177.b Dimension $8$ Group $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ Conductor $2.531\times 10^{13}$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $8$ Group: $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ Conductor: $$25311454040401$$$$\medspace = 2243^{4}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 9.1.25311454040401.1 Galois orbit size: $1$ Smallest permutation container: 12T177 Parity: even Projective image: $C_3^3:S_4$ Projective field: Galois closure of 9.1.25311454040401.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$: $$x^{3} + 9x + 92$$
Roots:
 $r_{ 1 }$ $=$ $$a + 17 + \left(12 a^{2} + 37 a + 81\right)\cdot 97 + \left(92 a^{2} + 21 a + 64\right)\cdot 97^{2} + \left(91 a^{2} + 50 a + 10\right)\cdot 97^{3} + \left(85 a^{2} + 67 a + 55\right)\cdot 97^{4} + \left(26 a^{2} + 93 a + 9\right)\cdot 97^{5} + \left(65 a^{2} + 96 a + 66\right)\cdot 97^{6} + \left(84 a^{2} + 45 a + 78\right)\cdot 97^{7} + \left(71 a^{2} + 45 a + 91\right)\cdot 97^{8} + \left(76 a^{2} + 38 a + 79\right)\cdot 97^{9} +O(97^{10})$$ a + 17 + (12*a^2 + 37*a + 81)*97 + (92*a^2 + 21*a + 64)*97^2 + (91*a^2 + 50*a + 10)*97^3 + (85*a^2 + 67*a + 55)*97^4 + (26*a^2 + 93*a + 9)*97^5 + (65*a^2 + 96*a + 66)*97^6 + (84*a^2 + 45*a + 78)*97^7 + (71*a^2 + 45*a + 91)*97^8 + (76*a^2 + 38*a + 79)*97^9+O(97^10) $r_{ 2 }$ $=$ $$13 a^{2} + 75 a + 95 + \left(91 a^{2} + 70 a + 70\right)\cdot 97 + \left(18 a^{2} + 61 a + 13\right)\cdot 97^{2} + \left(20 a^{2} + 31 a + 65\right)\cdot 97^{3} + \left(58 a^{2} + 61 a + 82\right)\cdot 97^{4} + \left(65 a^{2} + 46 a + 47\right)\cdot 97^{5} + \left(57 a^{2} + 24 a + 20\right)\cdot 97^{6} + \left(28 a^{2} + 35 a + 33\right)\cdot 97^{7} + \left(a^{2} + 27 a + 56\right)\cdot 97^{8} + \left(14 a^{2} + 81 a + 91\right)\cdot 97^{9} +O(97^{10})$$ 13*a^2 + 75*a + 95 + (91*a^2 + 70*a + 70)*97 + (18*a^2 + 61*a + 13)*97^2 + (20*a^2 + 31*a + 65)*97^3 + (58*a^2 + 61*a + 82)*97^4 + (65*a^2 + 46*a + 47)*97^5 + (57*a^2 + 24*a + 20)*97^6 + (28*a^2 + 35*a + 33)*97^7 + (a^2 + 27*a + 56)*97^8 + (14*a^2 + 81*a + 91)*97^9+O(97^10) $r_{ 3 }$ $=$ $$29 + 52\cdot 97 + 14\cdot 97^{2} + 7\cdot 97^{3} + 92\cdot 97^{4} + 91\cdot 97^{5} + 38\cdot 97^{6} + 80\cdot 97^{7} + 24\cdot 97^{8} + 20\cdot 97^{9} +O(97^{10})$$ 29 + 52*97 + 14*97^2 + 7*97^3 + 92*97^4 + 91*97^5 + 38*97^6 + 80*97^7 + 24*97^8 + 20*97^9+O(97^10) $r_{ 4 }$ $=$ $$88 a^{2} + 48 a + 52 + \left(79 a^{2} + 3 a + 38\right)\cdot 97 + \left(43 a^{2} + 78 a + 91\right)\cdot 97^{2} + \left(31 a^{2} + 18 a + 71\right)\cdot 97^{3} + \left(87 a^{2} + 2 a + 13\right)\cdot 97^{4} + \left(14 a^{2} + 27 a + 46\right)\cdot 97^{5} + \left(5 a^{2} + 71 a + 79\right)\cdot 97^{6} + \left(56 a^{2} + 45 a + 32\right)\cdot 97^{7} + \left(37 a^{2} + 46 a + 84\right)\cdot 97^{8} + \left(56 a^{2} + 81 a + 14\right)\cdot 97^{9} +O(97^{10})$$ 88*a^2 + 48*a + 52 + (79*a^2 + 3*a + 38)*97 + (43*a^2 + 78*a + 91)*97^2 + (31*a^2 + 18*a + 71)*97^3 + (87*a^2 + 2*a + 13)*97^4 + (14*a^2 + 27*a + 46)*97^5 + (5*a^2 + 71*a + 79)*97^6 + (56*a^2 + 45*a + 32)*97^7 + (37*a^2 + 46*a + 84)*97^8 + (56*a^2 + 81*a + 14)*97^9+O(97^10) $r_{ 5 }$ $=$ $$57 + 49\cdot 97 + 31\cdot 97^{2} + 91\cdot 97^{3} + 18\cdot 97^{4} + 97^{5} + 25\cdot 97^{6} + 95\cdot 97^{7} + 80\cdot 97^{8} + 70\cdot 97^{9} +O(97^{10})$$ 57 + 49*97 + 31*97^2 + 91*97^3 + 18*97^4 + 97^5 + 25*97^6 + 95*97^7 + 80*97^8 + 70*97^9+O(97^10) $r_{ 6 }$ $=$ $$47 a^{2} + 48 a + \left(74 a^{2} + 37 a + 6\right)\cdot 97 + \left(29 a^{2} + 46 a + 7\right)\cdot 97^{2} + \left(85 a^{2} + 46 a + 7\right)\cdot 97^{3} + \left(62 a^{2} + 14 a + 61\right)\cdot 97^{4} + \left(80 a^{2} + 93 a + 52\right)\cdot 97^{5} + \left(6 a^{2} + 36 a + 89\right)\cdot 97^{6} + \left(74 a^{2} + 73 a + 43\right)\cdot 97^{7} + \left(41 a^{2} + 11 a + 12\right)\cdot 97^{8} + \left(92 a^{2} + 74 a + 37\right)\cdot 97^{9} +O(97^{10})$$ 47*a^2 + 48*a + (74*a^2 + 37*a + 6)*97 + (29*a^2 + 46*a + 7)*97^2 + (85*a^2 + 46*a + 7)*97^3 + (62*a^2 + 14*a + 61)*97^4 + (80*a^2 + 93*a + 52)*97^5 + (6*a^2 + 36*a + 89)*97^6 + (74*a^2 + 73*a + 43)*97^7 + (41*a^2 + 11*a + 12)*97^8 + (92*a^2 + 74*a + 37)*97^9+O(97^10) $r_{ 7 }$ $=$ $$32 + 29\cdot 97 + 89\cdot 97^{2} + 31\cdot 97^{3} + 85\cdot 97^{4} + 6\cdot 97^{5} + 88\cdot 97^{6} + 82\cdot 97^{7} + 74\cdot 97^{8} + 80\cdot 97^{9} +O(97^{10})$$ 32 + 29*97 + 89*97^2 + 31*97^3 + 85*97^4 + 6*97^5 + 88*97^6 + 82*97^7 + 74*97^8 + 80*97^9+O(97^10) $r_{ 8 }$ $=$ $$84 a^{2} + 21 a + 36 + \left(90 a^{2} + 86 a + 69\right)\cdot 97 + \left(82 a^{2} + 13 a + 9\right)\cdot 97^{2} + \left(81 a^{2} + 15 a + 47\right)\cdot 97^{3} + \left(49 a^{2} + 65 a + 32\right)\cdot 97^{4} + \left(4 a^{2} + 53 a + 69\right)\cdot 97^{5} + \left(71 a^{2} + 72 a + 3\right)\cdot 97^{6} + \left(80 a^{2} + 15 a + 55\right)\cdot 97^{7} + \left(23 a^{2} + 24 a + 94\right)\cdot 97^{8} + \left(6 a^{2} + 74 a + 44\right)\cdot 97^{9} +O(97^{10})$$ 84*a^2 + 21*a + 36 + (90*a^2 + 86*a + 69)*97 + (82*a^2 + 13*a + 9)*97^2 + (81*a^2 + 15*a + 47)*97^3 + (49*a^2 + 65*a + 32)*97^4 + (4*a^2 + 53*a + 69)*97^5 + (71*a^2 + 72*a + 3)*97^6 + (80*a^2 + 15*a + 55)*97^7 + (23*a^2 + 24*a + 94)*97^8 + (6*a^2 + 74*a + 44)*97^9+O(97^10) $r_{ 9 }$ $=$ $$59 a^{2} + a + 72 + \left(39 a^{2} + 56 a + 87\right)\cdot 97 + \left(23 a^{2} + 69 a + 65\right)\cdot 97^{2} + \left(77 a^{2} + 31 a + 55\right)\cdot 97^{3} + \left(43 a^{2} + 80 a + 43\right)\cdot 97^{4} + \left(a^{2} + 73 a + 62\right)\cdot 97^{5} + \left(85 a^{2} + 85 a + 73\right)\cdot 97^{6} + \left(63 a^{2} + 74 a + 79\right)\cdot 97^{7} + \left(17 a^{2} + 38 a + 61\right)\cdot 97^{8} + \left(45 a^{2} + 38 a + 44\right)\cdot 97^{9} +O(97^{10})$$ 59*a^2 + a + 72 + (39*a^2 + 56*a + 87)*97 + (23*a^2 + 69*a + 65)*97^2 + (77*a^2 + 31*a + 55)*97^3 + (43*a^2 + 80*a + 43)*97^4 + (a^2 + 73*a + 62)*97^5 + (85*a^2 + 85*a + 73)*97^6 + (63*a^2 + 74*a + 79)*97^7 + (17*a^2 + 38*a + 61)*97^8 + (45*a^2 + 38*a + 44)*97^9+O(97^10)

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

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

### Character values on conjugacy classes

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