# Properties

 Label 56.259...744.105.a.a Dimension $56$ Group $A_8$ Conductor $2.599\times 10^{301}$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $56$ Group: $A_8$ Conductor: $$259\!\cdots\!744$$$$\medspace = 2^{202} \cdot 102953^{48}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: 8.0.319649416647163494229316963315979649024.1 Galois orbit size: $1$ Smallest permutation container: 105 Parity: even Determinant: 1.1.1t1.a.a Projective image: $A_8$ Projective stem field: 8.0.319649416647163494229316963315979649024.1

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - 28 x^{6} - 112 x^{5} - 210 x^{4} - 224 x^{3} - 140 x^{2} - 48 x + 823617$$  .

The roots of $f$ are computed in an extension of $\Q_{ 199 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 199 }$: $$x^{2} + 193 x + 3$$

Roots:
 $r_{ 1 }$ $=$ $$132 + 188\cdot 199 + 145\cdot 199^{2} + 128\cdot 199^{3} + 37\cdot 199^{4} + 151\cdot 199^{5} + 97\cdot 199^{6} + 39\cdot 199^{7} + 4\cdot 199^{8} + 192\cdot 199^{9} +O(199^{10})$$ $r_{ 2 }$ $=$ $$63 a + 49 + \left(84 a + 99\right)\cdot 199 + \left(39 a + 172\right)\cdot 199^{2} + \left(191 a + 121\right)\cdot 199^{3} + \left(76 a + 185\right)\cdot 199^{4} + \left(61 a + 21\right)\cdot 199^{5} + \left(16 a + 41\right)\cdot 199^{6} + \left(98 a + 30\right)\cdot 199^{7} + \left(33 a + 31\right)\cdot 199^{8} + \left(120 a + 3\right)\cdot 199^{9} +O(199^{10})$$ $r_{ 3 }$ $=$ $$115 + 9\cdot 199 + 157\cdot 199^{2} + 75\cdot 199^{3} + 77\cdot 199^{4} + 117\cdot 199^{5} + 48\cdot 199^{6} + 69\cdot 199^{7} + 37\cdot 199^{8} + 125\cdot 199^{9} +O(199^{10})$$ $r_{ 4 }$ $=$ $$42 a + 176 + \left(154 a + 63\right)\cdot 199 + \left(22 a + 175\right)\cdot 199^{2} + \left(173 a + 120\right)\cdot 199^{3} + \left(49 a + 142\right)\cdot 199^{4} + \left(6 a + 31\right)\cdot 199^{5} + \left(97 a + 85\right)\cdot 199^{6} + \left(70 a + 41\right)\cdot 199^{7} + \left(82 a + 131\right)\cdot 199^{8} + \left(114 a + 149\right)\cdot 199^{9} +O(199^{10})$$ $r_{ 5 }$ $=$ $$128 + 111\cdot 199 + 69\cdot 199^{2} + 26\cdot 199^{3} + 6\cdot 199^{4} + 157\cdot 199^{5} + 173\cdot 199^{6} + 159\cdot 199^{7} + 143\cdot 199^{8} + 102\cdot 199^{9} +O(199^{10})$$ $r_{ 6 }$ $=$ $$137 + 27\cdot 199 + 190\cdot 199^{2} + 144\cdot 199^{3} + 19\cdot 199^{4} + 183\cdot 199^{5} + 8\cdot 199^{6} + 83\cdot 199^{7} + 157\cdot 199^{8} + 171\cdot 199^{9} +O(199^{10})$$ $r_{ 7 }$ $=$ $$136 a + 29 + \left(114 a + 144\right)\cdot 199 + \left(159 a + 125\right)\cdot 199^{2} + \left(7 a + 35\right)\cdot 199^{3} + \left(122 a + 58\right)\cdot 199^{4} + \left(137 a + 114\right)\cdot 199^{5} + \left(182 a + 77\right)\cdot 199^{6} + \left(100 a + 5\right)\cdot 199^{7} + \left(165 a + 134\right)\cdot 199^{8} + \left(78 a + 93\right)\cdot 199^{9} +O(199^{10})$$ $r_{ 8 }$ $=$ $$157 a + 30 + \left(44 a + 151\right)\cdot 199 + \left(176 a + 157\right)\cdot 199^{2} + \left(25 a + 141\right)\cdot 199^{3} + \left(149 a + 69\right)\cdot 199^{4} + \left(192 a + 19\right)\cdot 199^{5} + \left(101 a + 64\right)\cdot 199^{6} + \left(128 a + 168\right)\cdot 199^{7} + \left(116 a + 156\right)\cdot 199^{8} + \left(84 a + 156\right)\cdot 199^{9} +O(199^{10})$$

## 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.