# Properties

 Label 45.782...528.336.a.a Dimension $45$ Group $A_8$ Conductor $7.827\times 10^{166}$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $45$ Group: $A_8$ Conductor: $$782\!\cdots\!528$$$$\medspace = 2^{100} \cdot 3217^{39}$$ Artin stem field: Galois closure of 8.0.72641749645773438449680384.1 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 8.0.72641749645773438449680384.1

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - 4x^{7} + 3217$$ x^8 - 4*x^7 + 3217 .

The roots of $f$ are computed in an extension of $\Q_{ 179 }$ to precision 8.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 179 }$: $$x^{2} + 172x + 2$$

Roots:
 $r_{ 1 }$ $=$ $$16 + 153\cdot 179 + 69\cdot 179^{2} + 79\cdot 179^{3} + 94\cdot 179^{4} + 38\cdot 179^{5} + 161\cdot 179^{6} + 108\cdot 179^{7} +O(179^{8})$$ 16 + 153*179 + 69*179^2 + 79*179^3 + 94*179^4 + 38*179^5 + 161*179^6 + 108*179^7+O(179^8) $r_{ 2 }$ $=$ $$137 + 154\cdot 179 + 67\cdot 179^{2} + 54\cdot 179^{3} + 108\cdot 179^{4} + 156\cdot 179^{5} + 26\cdot 179^{6} + 172\cdot 179^{7} +O(179^{8})$$ 137 + 154*179 + 67*179^2 + 54*179^3 + 108*179^4 + 156*179^5 + 26*179^6 + 172*179^7+O(179^8) $r_{ 3 }$ $=$ $$122 + 83\cdot 179 + 163\cdot 179^{2} + 12\cdot 179^{3} + 177\cdot 179^{4} + 156\cdot 179^{5} + 26\cdot 179^{6} + 39\cdot 179^{7} +O(179^{8})$$ 122 + 83*179 + 163*179^2 + 12*179^3 + 177*179^4 + 156*179^5 + 26*179^6 + 39*179^7+O(179^8) $r_{ 4 }$ $=$ $$137 a + 138 + \left(176 a + 107\right)\cdot 179 + \left(7 a + 119\right)\cdot 179^{2} + \left(113 a + 141\right)\cdot 179^{3} + 70\cdot 179^{4} + \left(90 a + 109\right)\cdot 179^{5} + \left(88 a + 127\right)\cdot 179^{6} + \left(20 a + 113\right)\cdot 179^{7} +O(179^{8})$$ 137*a + 138 + (176*a + 107)*179 + (7*a + 119)*179^2 + (113*a + 141)*179^3 + 70*179^4 + (90*a + 109)*179^5 + (88*a + 127)*179^6 + (20*a + 113)*179^7+O(179^8) $r_{ 5 }$ $=$ $$46 a + 71 + \left(31 a + 178\right)\cdot 179 + \left(54 a + 127\right)\cdot 179^{2} + \left(172 a + 112\right)\cdot 179^{3} + \left(102 a + 56\right)\cdot 179^{4} + \left(113 a + 176\right)\cdot 179^{5} + \left(78 a + 108\right)\cdot 179^{6} + \left(30 a + 100\right)\cdot 179^{7} +O(179^{8})$$ 46*a + 71 + (31*a + 178)*179 + (54*a + 127)*179^2 + (172*a + 112)*179^3 + (102*a + 56)*179^4 + (113*a + 176)*179^5 + (78*a + 108)*179^6 + (30*a + 100)*179^7+O(179^8) $r_{ 6 }$ $=$ $$178 + 89\cdot 179 + 50\cdot 179^{2} + 94\cdot 179^{3} + 178\cdot 179^{4} + 81\cdot 179^{5} + 135\cdot 179^{6} + 135\cdot 179^{7} +O(179^{8})$$ 178 + 89*179 + 50*179^2 + 94*179^3 + 178*179^4 + 81*179^5 + 135*179^6 + 135*179^7+O(179^8) $r_{ 7 }$ $=$ $$42 a + 23 + \left(2 a + 134\right)\cdot 179 + \left(171 a + 177\right)\cdot 179^{2} + \left(65 a + 29\right)\cdot 179^{3} + \left(178 a + 141\right)\cdot 179^{4} + \left(88 a + 22\right)\cdot 179^{5} + \left(90 a + 120\right)\cdot 179^{6} + \left(158 a + 168\right)\cdot 179^{7} +O(179^{8})$$ 42*a + 23 + (2*a + 134)*179 + (171*a + 177)*179^2 + (65*a + 29)*179^3 + (178*a + 141)*179^4 + (88*a + 22)*179^5 + (90*a + 120)*179^6 + (158*a + 168)*179^7+O(179^8) $r_{ 8 }$ $=$ $$133 a + 35 + \left(147 a + 172\right)\cdot 179 + \left(124 a + 117\right)\cdot 179^{2} + \left(6 a + 11\right)\cdot 179^{3} + \left(76 a + 68\right)\cdot 179^{4} + \left(65 a + 152\right)\cdot 179^{5} + \left(100 a + 8\right)\cdot 179^{6} + \left(148 a + 56\right)\cdot 179^{7} +O(179^{8})$$ 133*a + 35 + (147*a + 172)*179 + (124*a + 117)*179^2 + (6*a + 11)*179^3 + (76*a + 68)*179^4 + (65*a + 152)*179^5 + (100*a + 8)*179^6 + (148*a + 56)*179^7+O(179^8)

## 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.