# Properties

 Label 45.701...936.336.a.a Dimension $45$ Group $A_8$ Conductor $7.014\times 10^{269}$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $45$ Group: $A_8$ Conductor: $$701\!\cdots\!936$$$$\medspace = 2^{130} \cdot 11^{39} \cdot 74869^{39}$$ Artin stem field: Galois closure of 8.0.1277992348243533546275162547509832257536.1 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 8.0.1277992348243533546275162547509832257536.1

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - 28x^{6} - 112x^{5} - 210x^{4} - 224x^{3} - 140x^{2} - 48x + 823552$$ x^8 - 28*x^6 - 112*x^5 - 210*x^4 - 224*x^3 - 140*x^2 - 48*x + 823552 .

The roots of $f$ are computed in an extension of $\Q_{ 787 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 787 }$: $$x^{2} + 786x + 2$$

Roots:
 $r_{ 1 }$ $=$ $$215 a + 118 + \left(2 a + 494\right)\cdot 787 + \left(682 a + 42\right)\cdot 787^{2} + \left(634 a + 245\right)\cdot 787^{3} + \left(135 a + 160\right)\cdot 787^{4} + \left(232 a + 190\right)\cdot 787^{5} + \left(721 a + 681\right)\cdot 787^{6} + \left(148 a + 636\right)\cdot 787^{7} + \left(204 a + 757\right)\cdot 787^{8} + \left(112 a + 491\right)\cdot 787^{9} +O(787^{10})$$ 215*a + 118 + (2*a + 494)*787 + (682*a + 42)*787^2 + (634*a + 245)*787^3 + (135*a + 160)*787^4 + (232*a + 190)*787^5 + (721*a + 681)*787^6 + (148*a + 636)*787^7 + (204*a + 757)*787^8 + (112*a + 491)*787^9+O(787^10) $r_{ 2 }$ $=$ $$486 a + 233 + \left(280 a + 599\right)\cdot 787 + \left(656 a + 205\right)\cdot 787^{2} + \left(143 a + 44\right)\cdot 787^{3} + \left(552 a + 302\right)\cdot 787^{4} + \left(588 a + 569\right)\cdot 787^{5} + \left(746 a + 150\right)\cdot 787^{6} + \left(304 a + 338\right)\cdot 787^{7} + \left(359 a + 124\right)\cdot 787^{8} + \left(249 a + 666\right)\cdot 787^{9} +O(787^{10})$$ 486*a + 233 + (280*a + 599)*787 + (656*a + 205)*787^2 + (143*a + 44)*787^3 + (552*a + 302)*787^4 + (588*a + 569)*787^5 + (746*a + 150)*787^6 + (304*a + 338)*787^7 + (359*a + 124)*787^8 + (249*a + 666)*787^9+O(787^10) $r_{ 3 }$ $=$ $$637 a + 523 + \left(10 a + 20\right)\cdot 787 + \left(542 a + 560\right)\cdot 787^{2} + \left(134 a + 523\right)\cdot 787^{3} + \left(115 a + 52\right)\cdot 787^{4} + \left(440 a + 715\right)\cdot 787^{5} + \left(739 a + 9\right)\cdot 787^{6} + \left(33 a + 448\right)\cdot 787^{7} + \left(229 a + 709\right)\cdot 787^{8} + \left(165 a + 592\right)\cdot 787^{9} +O(787^{10})$$ 637*a + 523 + (10*a + 20)*787 + (542*a + 560)*787^2 + (134*a + 523)*787^3 + (115*a + 52)*787^4 + (440*a + 715)*787^5 + (739*a + 9)*787^6 + (33*a + 448)*787^7 + (229*a + 709)*787^8 + (165*a + 592)*787^9+O(787^10) $r_{ 4 }$ $=$ $$572 a + 333 + \left(784 a + 281\right)\cdot 787 + \left(104 a + 722\right)\cdot 787^{2} + \left(152 a + 197\right)\cdot 787^{3} + \left(651 a + 448\right)\cdot 787^{4} + \left(554 a + 286\right)\cdot 787^{5} + \left(65 a + 383\right)\cdot 787^{6} + \left(638 a + 64\right)\cdot 787^{7} + \left(582 a + 26\right)\cdot 787^{8} + \left(674 a + 400\right)\cdot 787^{9} +O(787^{10})$$ 572*a + 333 + (784*a + 281)*787 + (104*a + 722)*787^2 + (152*a + 197)*787^3 + (651*a + 448)*787^4 + (554*a + 286)*787^5 + (65*a + 383)*787^6 + (638*a + 64)*787^7 + (582*a + 26)*787^8 + (674*a + 400)*787^9+O(787^10) $r_{ 5 }$ $=$ $$394 a + 621 + \left(764 a + 9\right)\cdot 787 + \left(784 a + 749\right)\cdot 787^{2} + \left(780 a + 65\right)\cdot 787^{3} + \left(220 a + 607\right)\cdot 787^{4} + \left(254 a + 640\right)\cdot 787^{5} + \left(748 a + 11\right)\cdot 787^{6} + \left(315 a + 440\right)\cdot 787^{7} + \left(456 a + 546\right)\cdot 787^{8} + \left(152 a + 107\right)\cdot 787^{9} +O(787^{10})$$ 394*a + 621 + (764*a + 9)*787 + (784*a + 749)*787^2 + (780*a + 65)*787^3 + (220*a + 607)*787^4 + (254*a + 640)*787^5 + (748*a + 11)*787^6 + (315*a + 440)*787^7 + (456*a + 546)*787^8 + (152*a + 107)*787^9+O(787^10) $r_{ 6 }$ $=$ $$393 a + 228 + \left(22 a + 380\right)\cdot 787 + \left(2 a + 769\right)\cdot 787^{2} + \left(6 a + 61\right)\cdot 787^{3} + \left(566 a + 47\right)\cdot 787^{4} + \left(532 a + 674\right)\cdot 787^{5} + \left(38 a + 505\right)\cdot 787^{6} + \left(471 a + 7\right)\cdot 787^{7} + \left(330 a + 687\right)\cdot 787^{8} + \left(634 a + 590\right)\cdot 787^{9} +O(787^{10})$$ 393*a + 228 + (22*a + 380)*787 + (2*a + 769)*787^2 + (6*a + 61)*787^3 + (566*a + 47)*787^4 + (532*a + 674)*787^5 + (38*a + 505)*787^6 + (471*a + 7)*787^7 + (330*a + 687)*787^8 + (634*a + 590)*787^9+O(787^10) $r_{ 7 }$ $=$ $$150 a + 373 + \left(776 a + 181\right)\cdot 787 + \left(244 a + 304\right)\cdot 787^{2} + \left(652 a + 116\right)\cdot 787^{3} + \left(671 a + 33\right)\cdot 787^{4} + \left(346 a + 253\right)\cdot 787^{5} + \left(47 a + 309\right)\cdot 787^{6} + \left(753 a + 529\right)\cdot 787^{7} + \left(557 a + 117\right)\cdot 787^{8} + \left(621 a + 529\right)\cdot 787^{9} +O(787^{10})$$ 150*a + 373 + (776*a + 181)*787 + (244*a + 304)*787^2 + (652*a + 116)*787^3 + (671*a + 33)*787^4 + (346*a + 253)*787^5 + (47*a + 309)*787^6 + (753*a + 529)*787^7 + (557*a + 117)*787^8 + (621*a + 529)*787^9+O(787^10) $r_{ 8 }$ $=$ $$301 a + 719 + \left(506 a + 393\right)\cdot 787 + \left(130 a + 581\right)\cdot 787^{2} + \left(643 a + 318\right)\cdot 787^{3} + \left(234 a + 710\right)\cdot 787^{4} + \left(198 a + 605\right)\cdot 787^{5} + \left(40 a + 308\right)\cdot 787^{6} + \left(482 a + 683\right)\cdot 787^{7} + \left(427 a + 178\right)\cdot 787^{8} + \left(537 a + 556\right)\cdot 787^{9} +O(787^{10})$$ 301*a + 719 + (506*a + 393)*787 + (130*a + 581)*787^2 + (643*a + 318)*787^3 + (234*a + 710)*787^4 + (198*a + 605)*787^5 + (40*a + 308)*787^6 + (482*a + 683)*787^7 + (427*a + 178)*787^8 + (537*a + 556)*787^9+O(787^10)

## 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.