# Properties

 Label 56.106...784.105.a.a Dimension $56$ Group $A_8$ Conductor $1.060\times 10^{392}$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $56$ Group: $A_8$ Conductor: $$106\!\cdots\!784$$$$\medspace = 2^{144} \cdot 7^{70} \cdot 1075649^{48}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: 8.0.2340710530180884465731804242655471841228462751744.1 Galois orbit size: $1$ Smallest permutation container: 105 Parity: even Determinant: 1.1.1t1.a.a Projective image: $A_8$ Projective stem field: 8.0.2340710530180884465731804242655471841228462751744.1

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - 112 x^{6} - 896 x^{5} - 3360 x^{4} - 7168 x^{3} - 8960 x^{2} - 6144 x + 210825412$$  .

The roots of $f$ are computed in an extension of $\Q_{ 179 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 179 }$: $$x^{3} + 4 x + 177$$

Roots:
 $r_{ 1 }$ $=$ $$57 + 99\cdot 179 + 26\cdot 179^{2} + 6\cdot 179^{3} + 121\cdot 179^{4} + 61\cdot 179^{5} + 51\cdot 179^{6} + 21\cdot 179^{7} + 126\cdot 179^{8} + 71\cdot 179^{9} +O(179^{10})$$ $r_{ 2 }$ $=$ $$169 + 149\cdot 179 + 6\cdot 179^{2} + 77\cdot 179^{3} + 71\cdot 179^{4} + 154\cdot 179^{5} + 102\cdot 179^{6} + 83\cdot 179^{7} + 144\cdot 179^{8} + 160\cdot 179^{9} +O(179^{10})$$ $r_{ 3 }$ $=$ $$6 a^{2} + 148 a + 40 + \left(19 a^{2} + 66 a + 97\right)\cdot 179 + \left(88 a^{2} + 114 a + 128\right)\cdot 179^{2} + \left(154 a^{2} + 75 a + 16\right)\cdot 179^{3} + \left(131 a^{2} + 167 a + 86\right)\cdot 179^{4} + \left(34 a^{2} + 49 a + 7\right)\cdot 179^{5} + \left(69 a^{2} + 73 a + 120\right)\cdot 179^{6} + \left(101 a^{2} + 123 a + 152\right)\cdot 179^{7} + \left(56 a^{2} + 115 a + 32\right)\cdot 179^{8} + \left(165 a^{2} + 30 a + 55\right)\cdot 179^{9} +O(179^{10})$$ $r_{ 4 }$ $=$ $$25 a^{2} + 8 a + 27 + \left(147 a^{2} + 56 a + 24\right)\cdot 179 + \left(5 a^{2} + 37 a + 51\right)\cdot 179^{2} + \left(47 a^{2} + 135 a + 75\right)\cdot 179^{3} + \left(52 a^{2} + 78 a + 42\right)\cdot 179^{4} + \left(77 a^{2} + 127 a + 40\right)\cdot 179^{5} + \left(163 a^{2} + 133 a + 31\right)\cdot 179^{6} + \left(31 a^{2} + 81 a + 108\right)\cdot 179^{7} + \left(109 a^{2} + 89 a + 20\right)\cdot 179^{8} + \left(34 a^{2} + 16 a + 102\right)\cdot 179^{9} +O(179^{10})$$ $r_{ 5 }$ $=$ $$27 a^{2} + 12 a + 96 + \left(132 a^{2} + 22 a + 100\right)\cdot 179 + \left(62 a^{2} + 56 a + 120\right)\cdot 179^{2} + \left(2 a^{2} + 129 a + 28\right)\cdot 179^{3} + \left(131 a^{2} + 81 a + 24\right)\cdot 179^{4} + \left(131 a^{2} + 56 a + 87\right)\cdot 179^{5} + \left(163 a^{2} + 142 a + 133\right)\cdot 179^{6} + \left(72 a^{2} + 39 a + 76\right)\cdot 179^{7} + \left(121 a^{2} + 123 a + 86\right)\cdot 179^{8} + \left(66 a^{2} + 109 a + 90\right)\cdot 179^{9} +O(179^{10})$$ $r_{ 6 }$ $=$ $$76 a^{2} + 51 a + 163 + \left(166 a^{2} + 177 a + 15\right)\cdot 179 + \left(148 a^{2} + 12 a + 15\right)\cdot 179^{2} + \left(102 a^{2} + 94 a + 45\right)\cdot 179^{3} + \left(142 a^{2} + 40 a + 104\right)\cdot 179^{4} + \left(113 a^{2} + 27 a + 137\right)\cdot 179^{5} + \left(114 a^{2} + 138 a + 139\right)\cdot 179^{6} + \left(68 a^{2} + 44 a + 86\right)\cdot 179^{7} + \left(34 a^{2} + 118 a + 119\right)\cdot 179^{8} + \left(101 a^{2} + 33 a + 100\right)\cdot 179^{9} +O(179^{10})$$ $r_{ 7 }$ $=$ $$78 a^{2} + 120 a + 49 + \left(44 a^{2} + 124 a + 108\right)\cdot 179 + \left(24 a^{2} + 128 a + 40\right)\cdot 179^{2} + \left(29 a^{2} + 128 a + 87\right)\cdot 179^{3} + \left(163 a^{2} + 59 a + 99\right)\cdot 179^{4} + \left(166 a^{2} + 24 a + 40\right)\cdot 179^{5} + \left(79 a^{2} + 86 a + 47\right)\cdot 179^{6} + \left(78 a^{2} + 52 a + 53\right)\cdot 179^{7} + \left(35 a^{2} + 150 a + 122\right)\cdot 179^{8} + \left(43 a^{2} + 128 a + 5\right)\cdot 179^{9} +O(179^{10})$$ $r_{ 8 }$ $=$ $$146 a^{2} + 19 a + 115 + \left(27 a^{2} + 90 a + 120\right)\cdot 179 + \left(28 a^{2} + 8 a + 147\right)\cdot 179^{2} + \left(22 a^{2} + 153 a + 21\right)\cdot 179^{3} + \left(95 a^{2} + 108 a + 167\right)\cdot 179^{4} + \left(12 a^{2} + 72 a + 7\right)\cdot 179^{5} + \left(125 a^{2} + 142 a + 90\right)\cdot 179^{6} + \left(4 a^{2} + 15 a + 133\right)\cdot 179^{7} + \left(a^{2} + 119 a + 63\right)\cdot 179^{8} + \left(126 a^{2} + 38 a + 129\right)\cdot 179^{9} +O(179^{10})$$

## 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 $1$ $1$ $()$ $56$ $105$ $2$ $(1,2)(3,4)(5,6)(7,8)$ $8$ $210$ $2$ $(1,2)(3,4)$ $0$ $112$ $3$ $(1,2,3)$ $-4$ $1120$ $3$ $(1,2,3)(4,5,6)$ $-1$ $1260$ $4$ $(1,2,3,4)(5,6,7,8)$ $0$ $2520$ $4$ $(1,2,3,4)(5,6)$ $0$ $1344$ $5$ $(1,2,3,4,5)$ $1$ $1680$ $6$ $(1,2,3)(4,5)(6,7)$ $0$ $3360$ $6$ $(1,2,3,4,5,6)(7,8)$ $-1$ $2880$ $7$ $(1,2,3,4,5,6,7)$ $0$ $2880$ $7$ $(1,3,4,5,6,7,2)$ $0$ $1344$ $15$ $(1,2,3,4,5)(6,7,8)$ $1$ $1344$ $15$ $(1,3,4,5,2)(6,7,8)$ $1$

The blue line marks the conjugacy class containing complex conjugation.