Properties

Label 56.161...056.105.a.a
Dimension $56$
Group $A_8$
Conductor $1.619\times 10^{287}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $56$
Group: $A_8$
Conductor: \(161\!\cdots\!056\)\(\medspace = 2^{148} \cdot 29^{48} \cdot 3917^{48}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 8.0.2252730971538337484304305478905626624.1
Galois orbit size: $1$
Smallest permutation container: 105
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $A_8$
Projective stem field: 8.0.2252730971538337484304305478905626624.1

Defining polynomial

$f(x)$$=$\(x^{8} - 112 x^{6} - 896 x^{5} - 3360 x^{4} - 7168 x^{3} - 8960 x^{2} - 6144 x + 210826816\)  Toggle raw display.

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

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

Roots:
$r_{ 1 }$ $=$ \( 184 a + 142 + \left(180 a + 137\right)\cdot 199 + \left(192 a + 196\right)\cdot 199^{2} + \left(42 a + 173\right)\cdot 199^{3} + \left(21 a + 134\right)\cdot 199^{4} + \left(157 a + 87\right)\cdot 199^{5} + \left(179 a + 76\right)\cdot 199^{6} + \left(164 a + 79\right)\cdot 199^{7} + \left(65 a + 185\right)\cdot 199^{8} + \left(156 a + 130\right)\cdot 199^{9} +O(199^{10})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 23 a + 187 + \left(192 a + 72\right)\cdot 199 + \left(57 a + 161\right)\cdot 199^{2} + \left(5 a + 110\right)\cdot 199^{3} + \left(194 a + 24\right)\cdot 199^{4} + \left(194 a + 89\right)\cdot 199^{5} + \left(5 a + 130\right)\cdot 199^{6} + \left(96 a + 69\right)\cdot 199^{7} + \left(67 a + 92\right)\cdot 199^{8} + \left(71 a + 1\right)\cdot 199^{9} +O(199^{10})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 6 a + 192 + \left(143 a + 28\right)\cdot 199 + \left(154 a + 147\right)\cdot 199^{2} + \left(34 a + 183\right)\cdot 199^{3} + \left(146 a + 109\right)\cdot 199^{4} + \left(185 a + 165\right)\cdot 199^{5} + \left(25 a + 160\right)\cdot 199^{6} + \left(49 a + 48\right)\cdot 199^{7} + \left(128 a + 3\right)\cdot 199^{8} + \left(63 a + 68\right)\cdot 199^{9} +O(199^{10})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 183 a + 82 + \left(164 a + 104\right)\cdot 199 + \left(118 a + 152\right)\cdot 199^{2} + \left(131 a + 144\right)\cdot 199^{3} + \left(19 a + 87\right)\cdot 199^{4} + \left(143 a + 193\right)\cdot 199^{5} + \left(78 a + 195\right)\cdot 199^{6} + \left(88 a + 144\right)\cdot 199^{7} + \left(65 a + 31\right)\cdot 199^{8} + \left(68 a + 78\right)\cdot 199^{9} +O(199^{10})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 193 a + 29 + \left(55 a + 85\right)\cdot 199 + \left(44 a + 136\right)\cdot 199^{2} + \left(164 a + 38\right)\cdot 199^{3} + \left(52 a + 156\right)\cdot 199^{4} + \left(13 a + 138\right)\cdot 199^{5} + \left(173 a + 130\right)\cdot 199^{6} + \left(149 a + 118\right)\cdot 199^{7} + \left(70 a + 126\right)\cdot 199^{8} + \left(135 a + 122\right)\cdot 199^{9} +O(199^{10})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 16 a + 185 + \left(34 a + 114\right)\cdot 199 + \left(80 a + 103\right)\cdot 199^{2} + \left(67 a + 19\right)\cdot 199^{3} + \left(179 a + 74\right)\cdot 199^{4} + \left(55 a + 37\right)\cdot 199^{5} + \left(120 a + 127\right)\cdot 199^{6} + \left(110 a + 198\right)\cdot 199^{7} + \left(133 a + 136\right)\cdot 199^{8} + \left(130 a + 24\right)\cdot 199^{9} +O(199^{10})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 15 a + 52 + \left(18 a + 44\right)\cdot 199 + \left(6 a + 178\right)\cdot 199^{2} + \left(156 a + 39\right)\cdot 199^{3} + \left(177 a + 20\right)\cdot 199^{4} + \left(41 a + 14\right)\cdot 199^{5} + \left(19 a + 3\right)\cdot 199^{6} + \left(34 a + 93\right)\cdot 199^{7} + \left(133 a + 17\right)\cdot 199^{8} + \left(42 a + 8\right)\cdot 199^{9} +O(199^{10})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 176 a + 126 + \left(6 a + 8\right)\cdot 199 + \left(141 a + 118\right)\cdot 199^{2} + \left(193 a + 84\right)\cdot 199^{3} + \left(4 a + 188\right)\cdot 199^{4} + \left(4 a + 69\right)\cdot 199^{5} + \left(193 a + 170\right)\cdot 199^{6} + \left(102 a + 42\right)\cdot 199^{7} + \left(131 a + 3\right)\cdot 199^{8} + \left(127 a + 163\right)\cdot 199^{9} +O(199^{10})\)  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 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.