# Properties

 Label 70.298...616.120.a.a Dimension $70$ Group $A_8$ Conductor $2.985\times 10^{452}$ Root number $1$ Indicator $1$

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## Basic invariants

 Dimension: $70$ Group: $A_8$ Conductor: $$298\!\cdots\!616$$$$\medspace = 2^{204} \cdot 3294173^{60}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 8.0.1339918344154038594028110691225010840890507264.1 Galois orbit size: $1$ Smallest permutation container: 120 Parity: even Determinant: 1.1.1t1.a.a Projective image: $A_8$ Projective stem field: Galois closure of 8.0.1339918344154038594028110691225010840890507264.1

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - 112x^{6} - 896x^{5} - 3360x^{4} - 7168x^{3} - 8960x^{2} - 6144x + 210825280$$ x^8 - 112*x^6 - 896*x^5 - 3360*x^4 - 7168*x^3 - 8960*x^2 - 6144*x + 210825280 .

The roots of $f$ are computed in an extension of $\Q_{ 197 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 197 }$: $$x^{2} + 192x + 2$$

Roots:
 $r_{ 1 }$ $=$ $$59 a + 194 + \left(126 a + 186\right)\cdot 197 + \left(165 a + 86\right)\cdot 197^{2} + \left(51 a + 70\right)\cdot 197^{3} + \left(47 a + 171\right)\cdot 197^{4} + \left(65 a + 39\right)\cdot 197^{5} + \left(146 a + 134\right)\cdot 197^{6} + \left(25 a + 2\right)\cdot 197^{7} + 6 a\cdot 197^{8} + \left(35 a + 55\right)\cdot 197^{9} +O(197^{10})$$ 59*a + 194 + (126*a + 186)*197 + (165*a + 86)*197^2 + (51*a + 70)*197^3 + (47*a + 171)*197^4 + (65*a + 39)*197^5 + (146*a + 134)*197^6 + (25*a + 2)*197^7 + 6*a*197^8 + (35*a + 55)*197^9+O(197^10) $r_{ 2 }$ $=$ $$138 a + 95 + \left(70 a + 168\right)\cdot 197 + 31 a\cdot 197^{2} + \left(145 a + 164\right)\cdot 197^{3} + \left(149 a + 158\right)\cdot 197^{4} + \left(131 a + 121\right)\cdot 197^{5} + \left(50 a + 12\right)\cdot 197^{6} + \left(171 a + 182\right)\cdot 197^{7} + \left(190 a + 4\right)\cdot 197^{8} + \left(161 a + 27\right)\cdot 197^{9} +O(197^{10})$$ 138*a + 95 + (70*a + 168)*197 + 31*a*197^2 + (145*a + 164)*197^3 + (149*a + 158)*197^4 + (131*a + 121)*197^5 + (50*a + 12)*197^6 + (171*a + 182)*197^7 + (190*a + 4)*197^8 + (161*a + 27)*197^9+O(197^10) $r_{ 3 }$ $=$ $$111 a + 52 + \left(191 a + 109\right)\cdot 197 + \left(43 a + 4\right)\cdot 197^{2} + \left(155 a + 163\right)\cdot 197^{3} + \left(64 a + 78\right)\cdot 197^{4} + \left(117 a + 32\right)\cdot 197^{5} + \left(169 a + 146\right)\cdot 197^{6} + \left(96 a + 69\right)\cdot 197^{7} + \left(3 a + 148\right)\cdot 197^{8} + 52 a\cdot 197^{9} +O(197^{10})$$ 111*a + 52 + (191*a + 109)*197 + (43*a + 4)*197^2 + (155*a + 163)*197^3 + (64*a + 78)*197^4 + (117*a + 32)*197^5 + (169*a + 146)*197^6 + (96*a + 69)*197^7 + (3*a + 148)*197^8 + 52*a*197^9+O(197^10) $r_{ 4 }$ $=$ $$86 a + 16 + \left(5 a + 168\right)\cdot 197 + \left(153 a + 32\right)\cdot 197^{2} + \left(41 a + 107\right)\cdot 197^{3} + \left(132 a + 50\right)\cdot 197^{4} + \left(79 a + 160\right)\cdot 197^{5} + \left(27 a + 88\right)\cdot 197^{6} + \left(100 a + 187\right)\cdot 197^{7} + \left(193 a + 68\right)\cdot 197^{8} + \left(144 a + 60\right)\cdot 197^{9} +O(197^{10})$$ 86*a + 16 + (5*a + 168)*197 + (153*a + 32)*197^2 + (41*a + 107)*197^3 + (132*a + 50)*197^4 + (79*a + 160)*197^5 + (27*a + 88)*197^6 + (100*a + 187)*197^7 + (193*a + 68)*197^8 + (144*a + 60)*197^9+O(197^10) $r_{ 5 }$ $=$ $$76 a + 83 + \left(155 a + 143\right)\cdot 197 + \left(196 a + 151\right)\cdot 197^{2} + \left(184 a + 144\right)\cdot 197^{3} + \left(72 a + 167\right)\cdot 197^{4} + \left(70 a + 43\right)\cdot 197^{5} + \left(103 a + 157\right)\cdot 197^{6} + \left(42 a + 150\right)\cdot 197^{7} + \left(178 a + 184\right)\cdot 197^{8} + \left(17 a + 62\right)\cdot 197^{9} +O(197^{10})$$ 76*a + 83 + (155*a + 143)*197 + (196*a + 151)*197^2 + (184*a + 144)*197^3 + (72*a + 167)*197^4 + (70*a + 43)*197^5 + (103*a + 157)*197^6 + (42*a + 150)*197^7 + (178*a + 184)*197^8 + (17*a + 62)*197^9+O(197^10) $r_{ 6 }$ $=$ $$32 a + 158 + \left(53 a + 57\right)\cdot 197 + \left(77 a + 91\right)\cdot 197^{2} + \left(33 a + 80\right)\cdot 197^{3} + \left(74 a + 33\right)\cdot 197^{4} + \left(155 a + 76\right)\cdot 197^{5} + \left(183 a + 130\right)\cdot 197^{6} + \left(158 a + 154\right)\cdot 197^{7} + \left(7 a + 127\right)\cdot 197^{8} + \left(194 a + 19\right)\cdot 197^{9} +O(197^{10})$$ 32*a + 158 + (53*a + 57)*197 + (77*a + 91)*197^2 + (33*a + 80)*197^3 + (74*a + 33)*197^4 + (155*a + 76)*197^5 + (183*a + 130)*197^6 + (158*a + 154)*197^7 + (7*a + 127)*197^8 + (194*a + 19)*197^9+O(197^10) $r_{ 7 }$ $=$ $$165 a + 121 + \left(143 a + 94\right)\cdot 197 + \left(119 a + 30\right)\cdot 197^{2} + \left(163 a + 170\right)\cdot 197^{3} + \left(122 a + 173\right)\cdot 197^{4} + \left(41 a + 187\right)\cdot 197^{5} + \left(13 a + 105\right)\cdot 197^{6} + \left(38 a + 174\right)\cdot 197^{7} + \left(189 a + 7\right)\cdot 197^{8} + \left(2 a + 194\right)\cdot 197^{9} +O(197^{10})$$ 165*a + 121 + (143*a + 94)*197 + (119*a + 30)*197^2 + (163*a + 170)*197^3 + (122*a + 173)*197^4 + (41*a + 187)*197^5 + (13*a + 105)*197^6 + (38*a + 174)*197^7 + (189*a + 7)*197^8 + (2*a + 194)*197^9+O(197^10) $r_{ 8 }$ $=$ $$121 a + 69 + \left(41 a + 56\right)\cdot 197 + 192\cdot 197^{2} + \left(12 a + 84\right)\cdot 197^{3} + \left(124 a + 150\right)\cdot 197^{4} + \left(126 a + 125\right)\cdot 197^{5} + \left(93 a + 12\right)\cdot 197^{6} + \left(154 a + 63\right)\cdot 197^{7} + \left(18 a + 48\right)\cdot 197^{8} + \left(179 a + 171\right)\cdot 197^{9} +O(197^{10})$$ 121*a + 69 + (41*a + 56)*197 + 192*197^2 + (12*a + 84)*197^3 + (124*a + 150)*197^4 + (126*a + 125)*197^5 + (93*a + 12)*197^6 + (154*a + 63)*197^7 + (18*a + 48)*197^8 + (179*a + 171)*197^9+O(197^10)

## 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 $1$ $1$ $()$ $70$ $105$ $2$ $(1,2)(3,4)(5,6)(7,8)$ $-2$ $210$ $2$ $(1,2)(3,4)$ $2$ $112$ $3$ $(1,2,3)$ $-5$ $1120$ $3$ $(1,2,3)(4,5,6)$ $1$ $1260$ $4$ $(1,2,3,4)(5,6,7,8)$ $-2$ $2520$ $4$ $(1,2,3,4)(5,6)$ $0$ $1344$ $5$ $(1,2,3,4,5)$ $0$ $1680$ $6$ $(1,2,3)(4,5)(6,7)$ $-1$ $3360$ $6$ $(1,2,3,4,5,6)(7,8)$ $1$ $2880$ $7$ $(1,2,3,4,5,6,7)$ $0$ $2880$ $7$ $(1,3,4,5,6,7,2)$ $0$ $1344$ $15$ $(1,2,3,4,5)(6,7,8)$ $0$ $1344$ $15$ $(1,3,4,5,2)(6,7,8)$ $0$

The blue line marks the conjugacy class containing complex conjugation.