Properties

Label 56.514...696.105.a.a
Dimension $56$
Group $A_8$
Conductor $5.145\times 10^{388}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $56$
Group: $A_8$
Conductor: \(514\!\cdots\!696\)\(\medspace = 2^{156} \cdot 701^{48} \cdot 18797^{48}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 8.0.87813088530439533539051393535468530591915378212864.1
Galois orbit size: $1$
Smallest permutation container: 105
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $A_8$
Projective stem field: 8.0.87813088530439533539051393535468530591915378212864.1

Defining polynomial

$f(x)$$=$\(x^{8} - 112 x^{6} - 896 x^{5} - 3360 x^{4} - 7168 x^{3} - 8960 x^{2} - 6144 x + 210825360\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 593 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 593 }$: \(x^{2} + 592 x + 3\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 359 a + 470 + \left(268 a + 205\right)\cdot 593 + \left(125 a + 336\right)\cdot 593^{2} + \left(389 a + 184\right)\cdot 593^{3} + \left(531 a + 277\right)\cdot 593^{4} + \left(336 a + 455\right)\cdot 593^{5} + \left(209 a + 360\right)\cdot 593^{6} + \left(357 a + 54\right)\cdot 593^{7} + \left(333 a + 173\right)\cdot 593^{8} + \left(105 a + 59\right)\cdot 593^{9} +O(593^{10})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 234 a + 236 + \left(324 a + 115\right)\cdot 593 + \left(467 a + 193\right)\cdot 593^{2} + \left(203 a + 448\right)\cdot 593^{3} + \left(61 a + 419\right)\cdot 593^{4} + \left(256 a + 260\right)\cdot 593^{5} + \left(383 a + 233\right)\cdot 593^{6} + \left(235 a + 202\right)\cdot 593^{7} + \left(259 a + 149\right)\cdot 593^{8} + \left(487 a + 424\right)\cdot 593^{9} +O(593^{10})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 501 + 87\cdot 593 + 279\cdot 593^{2} + 419\cdot 593^{3} + 526\cdot 593^{4} + 192\cdot 593^{5} + 575\cdot 593^{6} + 95\cdot 593^{7} + 382\cdot 593^{8} + 256\cdot 593^{9} +O(593^{10})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 89 + 49\cdot 593 + 578\cdot 593^{2} + 3\cdot 593^{3} + 581\cdot 593^{4} + 27\cdot 593^{5} + 381\cdot 593^{6} + 535\cdot 593^{7} + 482\cdot 593^{8} + 190\cdot 593^{9} +O(593^{10})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 255 + 135\cdot 593 + 337\cdot 593^{2} + 590\cdot 593^{3} + 156\cdot 593^{4} + 22\cdot 593^{5} + 592\cdot 593^{6} + 248\cdot 593^{7} + 514\cdot 593^{8} + 460\cdot 593^{9} +O(593^{10})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 30 a + 205 + \left(262 a + 252\right)\cdot 593 + \left(510 a + 33\right)\cdot 593^{2} + \left(183 a + 191\right)\cdot 593^{3} + \left(266 a + 47\right)\cdot 593^{4} + \left(413 a + 411\right)\cdot 593^{5} + \left(491 a + 338\right)\cdot 593^{6} + \left(227 a + 413\right)\cdot 593^{7} + \left(431 a + 319\right)\cdot 593^{8} + \left(477 a + 340\right)\cdot 593^{9} +O(593^{10})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 381 + 448\cdot 593 + 332\cdot 593^{2} + 76\cdot 593^{3} + 233\cdot 593^{4} + 443\cdot 593^{5} + 66\cdot 593^{6} + 78\cdot 593^{7} + 420\cdot 593^{8} + 252\cdot 593^{9} +O(593^{10})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 563 a + 235 + \left(330 a + 484\right)\cdot 593 + \left(82 a + 281\right)\cdot 593^{2} + \left(409 a + 457\right)\cdot 593^{3} + \left(326 a + 129\right)\cdot 593^{4} + \left(179 a + 558\right)\cdot 593^{5} + \left(101 a + 416\right)\cdot 593^{6} + \left(365 a + 149\right)\cdot 593^{7} + \left(161 a + 523\right)\cdot 593^{8} + \left(115 a + 386\right)\cdot 593^{9} +O(593^{10})\)  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.