Properties

Label 56.205...264.105.a.a
Dimension $56$
Group $A_8$
Conductor $2.051\times 10^{330}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $56$
Group: $A_8$
Conductor: \(205\!\cdots\!264\)\(\medspace = 2^{154} \cdot 11^{48} \cdot 74869^{48}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 8.0.1277992348243533546275162547509832257536.1
Galois orbit size: $1$
Smallest permutation container: 105
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $A_8$
Projective stem field: 8.0.1277992348243533546275162547509832257536.1

Defining polynomial

$f(x)$$=$\(x^{8} - 28 x^{6} - 112 x^{5} - 210 x^{4} - 224 x^{3} - 140 x^{2} - 48 x + 823552\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 787 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 787 }$: \(x^{2} + 786 x + 2\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 215 a + 118 + \left(2 a + 494\right)\cdot 787 + \left(682 a + 42\right)\cdot 787^{2} + \left(634 a + 245\right)\cdot 787^{3} + \left(135 a + 160\right)\cdot 787^{4} + \left(232 a + 190\right)\cdot 787^{5} + \left(721 a + 681\right)\cdot 787^{6} + \left(148 a + 636\right)\cdot 787^{7} + \left(204 a + 757\right)\cdot 787^{8} + \left(112 a + 491\right)\cdot 787^{9} +O(787^{10})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 486 a + 233 + \left(280 a + 599\right)\cdot 787 + \left(656 a + 205\right)\cdot 787^{2} + \left(143 a + 44\right)\cdot 787^{3} + \left(552 a + 302\right)\cdot 787^{4} + \left(588 a + 569\right)\cdot 787^{5} + \left(746 a + 150\right)\cdot 787^{6} + \left(304 a + 338\right)\cdot 787^{7} + \left(359 a + 124\right)\cdot 787^{8} + \left(249 a + 666\right)\cdot 787^{9} +O(787^{10})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 637 a + 523 + \left(10 a + 20\right)\cdot 787 + \left(542 a + 560\right)\cdot 787^{2} + \left(134 a + 523\right)\cdot 787^{3} + \left(115 a + 52\right)\cdot 787^{4} + \left(440 a + 715\right)\cdot 787^{5} + \left(739 a + 9\right)\cdot 787^{6} + \left(33 a + 448\right)\cdot 787^{7} + \left(229 a + 709\right)\cdot 787^{8} + \left(165 a + 592\right)\cdot 787^{9} +O(787^{10})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 572 a + 333 + \left(784 a + 281\right)\cdot 787 + \left(104 a + 722\right)\cdot 787^{2} + \left(152 a + 197\right)\cdot 787^{3} + \left(651 a + 448\right)\cdot 787^{4} + \left(554 a + 286\right)\cdot 787^{5} + \left(65 a + 383\right)\cdot 787^{6} + \left(638 a + 64\right)\cdot 787^{7} + \left(582 a + 26\right)\cdot 787^{8} + \left(674 a + 400\right)\cdot 787^{9} +O(787^{10})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 394 a + 621 + \left(764 a + 9\right)\cdot 787 + \left(784 a + 749\right)\cdot 787^{2} + \left(780 a + 65\right)\cdot 787^{3} + \left(220 a + 607\right)\cdot 787^{4} + \left(254 a + 640\right)\cdot 787^{5} + \left(748 a + 11\right)\cdot 787^{6} + \left(315 a + 440\right)\cdot 787^{7} + \left(456 a + 546\right)\cdot 787^{8} + \left(152 a + 107\right)\cdot 787^{9} +O(787^{10})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 393 a + 228 + \left(22 a + 380\right)\cdot 787 + \left(2 a + 769\right)\cdot 787^{2} + \left(6 a + 61\right)\cdot 787^{3} + \left(566 a + 47\right)\cdot 787^{4} + \left(532 a + 674\right)\cdot 787^{5} + \left(38 a + 505\right)\cdot 787^{6} + \left(471 a + 7\right)\cdot 787^{7} + \left(330 a + 687\right)\cdot 787^{8} + \left(634 a + 590\right)\cdot 787^{9} +O(787^{10})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 150 a + 373 + \left(776 a + 181\right)\cdot 787 + \left(244 a + 304\right)\cdot 787^{2} + \left(652 a + 116\right)\cdot 787^{3} + \left(671 a + 33\right)\cdot 787^{4} + \left(346 a + 253\right)\cdot 787^{5} + \left(47 a + 309\right)\cdot 787^{6} + \left(753 a + 529\right)\cdot 787^{7} + \left(557 a + 117\right)\cdot 787^{8} + \left(621 a + 529\right)\cdot 787^{9} +O(787^{10})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 301 a + 719 + \left(506 a + 393\right)\cdot 787 + \left(130 a + 581\right)\cdot 787^{2} + \left(643 a + 318\right)\cdot 787^{3} + \left(234 a + 710\right)\cdot 787^{4} + \left(198 a + 605\right)\cdot 787^{5} + \left(40 a + 308\right)\cdot 787^{6} + \left(482 a + 683\right)\cdot 787^{7} + \left(427 a + 178\right)\cdot 787^{8} + \left(537 a + 556\right)\cdot 787^{9} +O(787^{10})\)  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.