Properties

Label 56.514...576.105.a.a
Dimension $56$
Group $A_8$
Conductor $5.146\times 10^{388}$
Root number $1$
Indicator $1$

Related objects

Learn more

Basic invariants

Dimension: $56$
Group: $A_8$
Conductor: \(514\!\cdots\!576\)\(\medspace = 2^{156} \cdot 23^{48} \cdot 572903^{48}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 8.0.87815967535129608031908549089667529769029306679296.1
Galois orbit size: $1$
Smallest permutation container: 105
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $A_8$
Projective stem field: 8.0.87815967535129608031908549089667529769029306679296.1

Defining polynomial

$f(x)$$=$\(x^{8} - 112 x^{6} - 896 x^{5} - 3360 x^{4} - 7168 x^{3} - 8960 x^{2} - 6144 x + 210826512\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 263 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 263 }$: \(x^{2} + 261 x + 5\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 210 + 104\cdot 263 + 86\cdot 263^{2} + 48\cdot 263^{3} + 214\cdot 263^{4} + 111\cdot 263^{5} + 131\cdot 263^{6} + 248\cdot 263^{7} + 193\cdot 263^{8} + 16\cdot 263^{9} +O(263^{10})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 75 a + 168 + \left(192 a + 132\right)\cdot 263 + \left(217 a + 99\right)\cdot 263^{2} + \left(35 a + 65\right)\cdot 263^{3} + \left(141 a + 160\right)\cdot 263^{4} + \left(211 a + 25\right)\cdot 263^{5} + \left(126 a + 103\right)\cdot 263^{6} + \left(170 a + 99\right)\cdot 263^{7} + \left(142 a + 158\right)\cdot 263^{8} + \left(249 a + 136\right)\cdot 263^{9} +O(263^{10})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 45 + 43\cdot 263 + 214\cdot 263^{2} + 253\cdot 263^{3} + 80\cdot 263^{4} + 21\cdot 263^{5} + 169\cdot 263^{6} + 147\cdot 263^{7} + 211\cdot 263^{8} + 189\cdot 263^{9} +O(263^{10})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 179 a + 69 + \left(43 a + 73\right)\cdot 263 + \left(260 a + 36\right)\cdot 263^{2} + \left(232 a + 184\right)\cdot 263^{3} + \left(131 a + 158\right)\cdot 263^{4} + \left(166 a + 99\right)\cdot 263^{5} + \left(134 a + 57\right)\cdot 263^{6} + \left(255 a + 236\right)\cdot 263^{7} + \left(107 a + 146\right)\cdot 263^{8} + \left(68 a + 17\right)\cdot 263^{9} +O(263^{10})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 84 a + 164 + \left(219 a + 244\right)\cdot 263 + \left(2 a + 249\right)\cdot 263^{2} + \left(30 a + 126\right)\cdot 263^{3} + \left(131 a + 189\right)\cdot 263^{4} + \left(96 a + 37\right)\cdot 263^{5} + \left(128 a + 160\right)\cdot 263^{6} + \left(7 a + 86\right)\cdot 263^{7} + \left(155 a + 107\right)\cdot 263^{8} + \left(194 a + 46\right)\cdot 263^{9} +O(263^{10})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 184 + 217\cdot 263 + 6\cdot 263^{2} + 92\cdot 263^{3} + 48\cdot 263^{4} + 92\cdot 263^{5} + 262\cdot 263^{6} + 65\cdot 263^{7} + 47\cdot 263^{8} + 2\cdot 263^{9} +O(263^{10})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 188 a + 55 + \left(70 a + 179\right)\cdot 263 + \left(45 a + 79\right)\cdot 263^{2} + \left(227 a + 182\right)\cdot 263^{3} + \left(121 a + 143\right)\cdot 263^{4} + \left(51 a + 44\right)\cdot 263^{5} + \left(136 a + 145\right)\cdot 263^{6} + \left(92 a + 50\right)\cdot 263^{7} + \left(120 a + 10\right)\cdot 263^{8} + \left(13 a + 230\right)\cdot 263^{9} +O(263^{10})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 157 + 56\cdot 263 + 16\cdot 263^{2} + 99\cdot 263^{3} + 56\cdot 263^{4} + 93\cdot 263^{5} + 23\cdot 263^{6} + 117\cdot 263^{7} + 176\cdot 263^{8} + 149\cdot 263^{9} +O(263^{10})\)  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.