# Properties

 Label 45.761...136.336.a.a Dimension $45$ Group $A_8$ Conductor $7.610\times 10^{229}$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $45$ Group: $A_8$ Conductor: $$761\!\cdots\!136$$$$\medspace = 2^{176} \cdot 7^{56} \cdot 11^{39} \cdot 191^{39}$$ Artin stem field: Galois closure of 8.0.133100753213221593424899389161209856.1 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 8.0.133100753213221593424899389161209856.1

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - 28x^{6} - 112x^{5} - 210x^{4} - 224x^{3} - 140x^{2} - 48x + 823585$$ x^8 - 28*x^6 - 112*x^5 - 210*x^4 - 224*x^3 - 140*x^2 - 48*x + 823585 .

The roots of $f$ are computed in an extension of $\Q_{ 193 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 193 }$: $$x^{2} + 192x + 5$$

Roots:
 $r_{ 1 }$ $=$ $$121 a + 67 + \left(178 a + 135\right)\cdot 193 + \left(181 a + 110\right)\cdot 193^{2} + \left(150 a + 160\right)\cdot 193^{3} + \left(66 a + 82\right)\cdot 193^{4} + \left(138 a + 189\right)\cdot 193^{5} + \left(151 a + 164\right)\cdot 193^{6} + \left(46 a + 192\right)\cdot 193^{7} + \left(114 a + 92\right)\cdot 193^{8} + \left(140 a + 174\right)\cdot 193^{9} +O(193^{10})$$ 121*a + 67 + (178*a + 135)*193 + (181*a + 110)*193^2 + (150*a + 160)*193^3 + (66*a + 82)*193^4 + (138*a + 189)*193^5 + (151*a + 164)*193^6 + (46*a + 192)*193^7 + (114*a + 92)*193^8 + (140*a + 174)*193^9+O(193^10) $r_{ 2 }$ $=$ $$76 + 117\cdot 193 + 102\cdot 193^{2} + 88\cdot 193^{3} + 74\cdot 193^{4} + 103\cdot 193^{5} + 84\cdot 193^{6} + 153\cdot 193^{7} + 141\cdot 193^{8} + 91\cdot 193^{9} +O(193^{10})$$ 76 + 117*193 + 102*193^2 + 88*193^3 + 74*193^4 + 103*193^5 + 84*193^6 + 153*193^7 + 141*193^8 + 91*193^9+O(193^10) $r_{ 3 }$ $=$ $$11 a + 191 + \left(162 a + 14\right)\cdot 193 + \left(68 a + 131\right)\cdot 193^{2} + \left(114 a + 104\right)\cdot 193^{3} + \left(154 a + 71\right)\cdot 193^{4} + \left(133 a + 166\right)\cdot 193^{5} + \left(132 a + 13\right)\cdot 193^{6} + \left(173 a + 85\right)\cdot 193^{7} + \left(21 a + 87\right)\cdot 193^{8} + \left(98 a + 104\right)\cdot 193^{9} +O(193^{10})$$ 11*a + 191 + (162*a + 14)*193 + (68*a + 131)*193^2 + (114*a + 104)*193^3 + (154*a + 71)*193^4 + (133*a + 166)*193^5 + (132*a + 13)*193^6 + (173*a + 85)*193^7 + (21*a + 87)*193^8 + (98*a + 104)*193^9+O(193^10) $r_{ 4 }$ $=$ $$114 + 166\cdot 193 + 29\cdot 193^{2} + 89\cdot 193^{3} + 134\cdot 193^{4} + 74\cdot 193^{5} + 150\cdot 193^{6} + 70\cdot 193^{7} + 65\cdot 193^{8} + 5\cdot 193^{9} +O(193^{10})$$ 114 + 166*193 + 29*193^2 + 89*193^3 + 134*193^4 + 74*193^5 + 150*193^6 + 70*193^7 + 65*193^8 + 5*193^9+O(193^10) $r_{ 5 }$ $=$ $$182 a + 9 + \left(30 a + 166\right)\cdot 193 + \left(124 a + 37\right)\cdot 193^{2} + \left(78 a + 150\right)\cdot 193^{3} + \left(38 a + 111\right)\cdot 193^{4} + \left(59 a + 145\right)\cdot 193^{5} + \left(60 a + 12\right)\cdot 193^{6} + \left(19 a + 126\right)\cdot 193^{7} + \left(171 a + 128\right)\cdot 193^{8} + \left(94 a + 180\right)\cdot 193^{9} +O(193^{10})$$ 182*a + 9 + (30*a + 166)*193 + (124*a + 37)*193^2 + (78*a + 150)*193^3 + (38*a + 111)*193^4 + (59*a + 145)*193^5 + (60*a + 12)*193^6 + (19*a + 126)*193^7 + (171*a + 128)*193^8 + (94*a + 180)*193^9+O(193^10) $r_{ 6 }$ $=$ $$95 + 15\cdot 193 + 126\cdot 193^{2} + 150\cdot 193^{3} + 70\cdot 193^{4} + 162\cdot 193^{5} + 41\cdot 193^{6} + 48\cdot 193^{7} + 121\cdot 193^{8} + 17\cdot 193^{9} +O(193^{10})$$ 95 + 15*193 + 126*193^2 + 150*193^3 + 70*193^4 + 162*193^5 + 41*193^6 + 48*193^7 + 121*193^8 + 17*193^9+O(193^10) $r_{ 7 }$ $=$ $$72 a + 188 + \left(14 a + 192\right)\cdot 193 + \left(11 a + 113\right)\cdot 193^{2} + \left(42 a + 129\right)\cdot 193^{3} + \left(126 a + 191\right)\cdot 193^{4} + \left(54 a + 67\right)\cdot 193^{5} + \left(41 a + 178\right)\cdot 193^{6} + \left(146 a + 87\right)\cdot 193^{7} + \left(78 a + 160\right)\cdot 193^{8} + \left(52 a + 7\right)\cdot 193^{9} +O(193^{10})$$ 72*a + 188 + (14*a + 192)*193 + (11*a + 113)*193^2 + (42*a + 129)*193^3 + (126*a + 191)*193^4 + (54*a + 67)*193^5 + (41*a + 178)*193^6 + (146*a + 87)*193^7 + (78*a + 160)*193^8 + (52*a + 7)*193^9+O(193^10) $r_{ 8 }$ $=$ $$32 + 156\cdot 193 + 119\cdot 193^{2} + 91\cdot 193^{3} + 34\cdot 193^{4} + 55\cdot 193^{5} + 125\cdot 193^{6} + 7\cdot 193^{7} + 167\cdot 193^{8} + 189\cdot 193^{9} +O(193^{10})$$ 32 + 156*193 + 119*193^2 + 91*193^3 + 34*193^4 + 55*193^5 + 125*193^6 + 7*193^7 + 167*193^8 + 189*193^9+O(193^10)

## 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 $1$ $1$ $()$ $45$ $105$ $2$ $(1,2)(3,4)(5,6)(7,8)$ $-3$ $210$ $2$ $(1,2)(3,4)$ $-3$ $112$ $3$ $(1,2,3)$ $0$ $1120$ $3$ $(1,2,3)(4,5,6)$ $0$ $1260$ $4$ $(1,2,3,4)(5,6,7,8)$ $1$ $2520$ $4$ $(1,2,3,4)(5,6)$ $1$ $1344$ $5$ $(1,2,3,4,5)$ $0$ $1680$ $6$ $(1,2,3)(4,5)(6,7)$ $0$ $3360$ $6$ $(1,2,3,4,5,6)(7,8)$ $0$ $2880$ $7$ $(1,2,3,4,5,6,7)$ $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ $2880$ $7$ $(1,3,4,5,6,7,2)$ $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ $1344$ $15$ $(1,2,3,4,5)(6,7,8)$ $0$ $1344$ $15$ $(1,3,4,5,2)(6,7,8)$ $0$

The blue line marks the conjugacy class containing complex conjugation.