# Properties

 Label 56.276...296.105.a Dimension $56$ Group $A_8$ Conductor $2.763\times 10^{279}$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $56$ Group: $A_8$ Conductor: $$276\!\cdots\!296$$$$\medspace = 2^{202} \cdot 7^{70} \cdot 11^{48} \cdot 191^{48}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 8.0.133100753213221593424899389161209856.1 Galois orbit size: $1$ Smallest permutation container: 105 Parity: even Projective image: $A_8$ Projective field: 8.0.133100753213221593424899389161209856.1

## Galois action

### Roots of defining polynomial

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

### 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 values $c1$ $1$ $1$ $()$ $56$ $105$ $2$ $(1,2)(3,4)(5,6)(7,8)$ $8$ $210$ $2$ $(1,2)(3,4)$ $0$ $112$ $3$ $(1,2,3)$ $-4$ $1120$ $3$ $(1,2,3)(4,5,6)$ $-1$ $1260$ $4$ $(1,2,3,4)(5,6,7,8)$ $0$ $2520$ $4$ $(1,2,3,4)(5,6)$ $0$ $1344$ $5$ $(1,2,3,4,5)$ $1$ $1680$ $6$ $(1,2,3)(4,5)(6,7)$ $0$ $3360$ $6$ $(1,2,3,4,5,6)(7,8)$ $-1$ $2880$ $7$ $(1,2,3,4,5,6,7)$ $0$ $2880$ $7$ $(1,3,4,5,6,7,2)$ $0$ $1344$ $15$ $(1,2,3,4,5)(6,7,8)$ $1$ $1344$ $15$ $(1,3,4,5,2)(6,7,8)$ $1$
The blue line marks the conjugacy class containing complex conjugation.