# Properties

 Label 45.296...752.336.a.b Dimension $45$ Group $A_8$ Conductor $2.969\times 10^{248}$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $45$ Group: $A_8$ Conductor: $$296\!\cdots\!752$$$$\medspace = 2^{176} \cdot 113^{39} \cdot 911^{39}$$ Artin stem field: Galois closure of 8.0.319463173328482073097337827900516204544.1 Galois orbit size: $2$ Smallest permutation container: 336 Parity: even Determinant: 1.1.1t1.a.a Projective image: $A_8$ Projective stem field: Galois closure of 8.0.319463173328482073097337827900516204544.1

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - 28x^{6} - 112x^{5} - 210x^{4} - 224x^{3} - 140x^{2} - 48x + 823537$$ x^8 - 28*x^6 - 112*x^5 - 210*x^4 - 224*x^3 - 140*x^2 - 48*x + 823537 .

The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$: $$x^{2} + 96x + 5$$

Roots:
 $r_{ 1 }$ $=$ $$4 a + 6 + \left(41 a + 10\right)\cdot 97 + \left(23 a + 12\right)\cdot 97^{2} + \left(79 a + 5\right)\cdot 97^{3} + 23 a\cdot 97^{4} + \left(6 a + 6\right)\cdot 97^{5} + \left(92 a + 73\right)\cdot 97^{6} + \left(65 a + 24\right)\cdot 97^{7} + \left(55 a + 33\right)\cdot 97^{8} + \left(32 a + 6\right)\cdot 97^{9} +O(97^{10})$$ 4*a + 6 + (41*a + 10)*97 + (23*a + 12)*97^2 + (79*a + 5)*97^3 + 23*a*97^4 + (6*a + 6)*97^5 + (92*a + 73)*97^6 + (65*a + 24)*97^7 + (55*a + 33)*97^8 + (32*a + 6)*97^9+O(97^10) $r_{ 2 }$ $=$ $$92 a + 35 + \left(49 a + 49\right)\cdot 97 + \left(74 a + 5\right)\cdot 97^{2} + \left(53 a + 94\right)\cdot 97^{3} + \left(42 a + 96\right)\cdot 97^{4} + \left(61 a + 4\right)\cdot 97^{5} + \left(5 a + 86\right)\cdot 97^{6} + \left(49 a + 87\right)\cdot 97^{7} + \left(63 a + 84\right)\cdot 97^{8} + \left(21 a + 41\right)\cdot 97^{9} +O(97^{10})$$ 92*a + 35 + (49*a + 49)*97 + (74*a + 5)*97^2 + (53*a + 94)*97^3 + (42*a + 96)*97^4 + (61*a + 4)*97^5 + (5*a + 86)*97^6 + (49*a + 87)*97^7 + (63*a + 84)*97^8 + (21*a + 41)*97^9+O(97^10) $r_{ 3 }$ $=$ $$70 a + 22 + \left(3 a + 3\right)\cdot 97 + \left(87 a + 21\right)\cdot 97^{2} + \left(61 a + 63\right)\cdot 97^{3} + \left(65 a + 13\right)\cdot 97^{4} + \left(71 a + 73\right)\cdot 97^{5} + \left(13 a + 44\right)\cdot 97^{6} + \left(46 a + 89\right)\cdot 97^{7} + \left(84 a + 61\right)\cdot 97^{8} + \left(14 a + 89\right)\cdot 97^{9} +O(97^{10})$$ 70*a + 22 + (3*a + 3)*97 + (87*a + 21)*97^2 + (61*a + 63)*97^3 + (65*a + 13)*97^4 + (71*a + 73)*97^5 + (13*a + 44)*97^6 + (46*a + 89)*97^7 + (84*a + 61)*97^8 + (14*a + 89)*97^9+O(97^10) $r_{ 4 }$ $=$ $$85 a + 54 + \left(55 a + 84\right)\cdot 97 + \left(45 a + 66\right)\cdot 97^{2} + \left(81 a + 8\right)\cdot 97^{3} + \left(10 a + 53\right)\cdot 97^{4} + \left(87 a + 19\right)\cdot 97^{5} + \left(6 a + 44\right)\cdot 97^{6} + \left(45 a + 93\right)\cdot 97^{7} + \left(33 a + 46\right)\cdot 97^{8} + \left(80 a + 51\right)\cdot 97^{9} +O(97^{10})$$ 85*a + 54 + (55*a + 84)*97 + (45*a + 66)*97^2 + (81*a + 8)*97^3 + (10*a + 53)*97^4 + (87*a + 19)*97^5 + (6*a + 44)*97^6 + (45*a + 93)*97^7 + (33*a + 46)*97^8 + (80*a + 51)*97^9+O(97^10) $r_{ 5 }$ $=$ $$5 a + 30 + \left(47 a + 7\right)\cdot 97 + \left(22 a + 30\right)\cdot 97^{2} + \left(43 a + 73\right)\cdot 97^{3} + \left(54 a + 85\right)\cdot 97^{4} + \left(35 a + 23\right)\cdot 97^{5} + \left(91 a + 30\right)\cdot 97^{6} + \left(47 a + 34\right)\cdot 97^{7} + \left(33 a + 2\right)\cdot 97^{8} + 75 a\cdot 97^{9} +O(97^{10})$$ 5*a + 30 + (47*a + 7)*97 + (22*a + 30)*97^2 + (43*a + 73)*97^3 + (54*a + 85)*97^4 + (35*a + 23)*97^5 + (91*a + 30)*97^6 + (47*a + 34)*97^7 + (33*a + 2)*97^8 + 75*a*97^9+O(97^10) $r_{ 6 }$ $=$ $$27 a + 92 + \left(93 a + 33\right)\cdot 97 + \left(9 a + 7\right)\cdot 97^{2} + \left(35 a + 38\right)\cdot 97^{3} + \left(31 a + 17\right)\cdot 97^{4} + \left(25 a + 79\right)\cdot 97^{5} + \left(83 a + 83\right)\cdot 97^{6} + \left(50 a + 24\right)\cdot 97^{7} + \left(12 a + 3\right)\cdot 97^{8} + \left(82 a + 20\right)\cdot 97^{9} +O(97^{10})$$ 27*a + 92 + (93*a + 33)*97 + (9*a + 7)*97^2 + (35*a + 38)*97^3 + (31*a + 17)*97^4 + (25*a + 79)*97^5 + (83*a + 83)*97^6 + (50*a + 24)*97^7 + (12*a + 3)*97^8 + (82*a + 20)*97^9+O(97^10) $r_{ 7 }$ $=$ $$12 a + 42 + \left(41 a + 55\right)\cdot 97 + \left(51 a + 56\right)\cdot 97^{2} + \left(15 a + 44\right)\cdot 97^{3} + \left(86 a + 79\right)\cdot 97^{4} + \left(9 a + 95\right)\cdot 97^{5} + \left(90 a + 60\right)\cdot 97^{6} + \left(51 a + 34\right)\cdot 97^{7} + \left(63 a + 35\right)\cdot 97^{8} + \left(16 a + 1\right)\cdot 97^{9} +O(97^{10})$$ 12*a + 42 + (41*a + 55)*97 + (51*a + 56)*97^2 + (15*a + 44)*97^3 + (86*a + 79)*97^4 + (9*a + 95)*97^5 + (90*a + 60)*97^6 + (51*a + 34)*97^7 + (63*a + 35)*97^8 + (16*a + 1)*97^9+O(97^10) $r_{ 8 }$ $=$ $$93 a + 10 + \left(55 a + 47\right)\cdot 97 + \left(73 a + 91\right)\cdot 97^{2} + \left(17 a + 60\right)\cdot 97^{3} + \left(73 a + 41\right)\cdot 97^{4} + \left(90 a + 85\right)\cdot 97^{5} + \left(4 a + 61\right)\cdot 97^{6} + \left(31 a + 95\right)\cdot 97^{7} + \left(41 a + 22\right)\cdot 97^{8} + \left(64 a + 80\right)\cdot 97^{9} +O(97^{10})$$ 93*a + 10 + (55*a + 47)*97 + (73*a + 91)*97^2 + (17*a + 60)*97^3 + (73*a + 41)*97^4 + (90*a + 85)*97^5 + (4*a + 61)*97^6 + (31*a + 95)*97^7 + (41*a + 22)*97^8 + (64*a + 80)*97^9+O(97^10)

## Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character value $1$ $1$ $()$ $45$ $105$ $2$ $(1,2)(3,4)(5,6)(7,8)$ $-3$ $210$ $2$ $(1,2)(3,4)$ $-3$ $112$ $3$ $(1,2,3)$ $0$ $1120$ $3$ $(1,2,3)(4,5,6)$ $0$ $1260$ $4$ $(1,2,3,4)(5,6,7,8)$ $1$ $2520$ $4$ $(1,2,3,4)(5,6)$ $1$ $1344$ $5$ $(1,2,3,4,5)$ $0$ $1680$ $6$ $(1,2,3)(4,5)(6,7)$ $0$ $3360$ $6$ $(1,2,3,4,5,6)(7,8)$ $0$ $2880$ $7$ $(1,2,3,4,5,6,7)$ $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ $2880$ $7$ $(1,3,4,5,6,7,2)$ $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ $1344$ $15$ $(1,2,3,4,5)(6,7,8)$ $0$ $1344$ $15$ $(1,3,4,5,2)(6,7,8)$ $0$

The blue line marks the conjugacy class containing complex conjugation.