Properties

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

Related objects

Learn more

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

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

SizeOrderAction 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.