Properties

Label 56.258...504.105.a.a
Dimension $56$
Group $A_8$
Conductor $2.586\times 10^{301}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $56$
Group: $A_8$
Conductor: \(258\!\cdots\!504\)\(\medspace = 2^{202} \cdot 113^{48} \cdot 911^{48}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 8.0.319463173328482073097337827900516204544.1
Galois orbit size: $1$
Smallest permutation container: 105
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $A_8$
Projective stem field: 8.0.319463173328482073097337827900516204544.1

Defining polynomial

$f(x)$$=$\(x^{8} - 28 x^{6} - 112 x^{5} - 210 x^{4} - 224 x^{3} - 140 x^{2} - 48 x + 823537\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$: \(x^{2} + 96 x + 5\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 4 a + 6 + \left(41 a + 10\right)\cdot 97 + \left(23 a + 12\right)\cdot 97^{2} + \left(79 a + 5\right)\cdot 97^{3} + 23 a\cdot 97^{4} + \left(6 a + 6\right)\cdot 97^{5} + \left(92 a + 73\right)\cdot 97^{6} + \left(65 a + 24\right)\cdot 97^{7} + \left(55 a + 33\right)\cdot 97^{8} + \left(32 a + 6\right)\cdot 97^{9} +O(97^{10})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 92 a + 35 + \left(49 a + 49\right)\cdot 97 + \left(74 a + 5\right)\cdot 97^{2} + \left(53 a + 94\right)\cdot 97^{3} + \left(42 a + 96\right)\cdot 97^{4} + \left(61 a + 4\right)\cdot 97^{5} + \left(5 a + 86\right)\cdot 97^{6} + \left(49 a + 87\right)\cdot 97^{7} + \left(63 a + 84\right)\cdot 97^{8} + \left(21 a + 41\right)\cdot 97^{9} +O(97^{10})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 70 a + 22 + \left(3 a + 3\right)\cdot 97 + \left(87 a + 21\right)\cdot 97^{2} + \left(61 a + 63\right)\cdot 97^{3} + \left(65 a + 13\right)\cdot 97^{4} + \left(71 a + 73\right)\cdot 97^{5} + \left(13 a + 44\right)\cdot 97^{6} + \left(46 a + 89\right)\cdot 97^{7} + \left(84 a + 61\right)\cdot 97^{8} + \left(14 a + 89\right)\cdot 97^{9} +O(97^{10})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 85 a + 54 + \left(55 a + 84\right)\cdot 97 + \left(45 a + 66\right)\cdot 97^{2} + \left(81 a + 8\right)\cdot 97^{3} + \left(10 a + 53\right)\cdot 97^{4} + \left(87 a + 19\right)\cdot 97^{5} + \left(6 a + 44\right)\cdot 97^{6} + \left(45 a + 93\right)\cdot 97^{7} + \left(33 a + 46\right)\cdot 97^{8} + \left(80 a + 51\right)\cdot 97^{9} +O(97^{10})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 5 a + 30 + \left(47 a + 7\right)\cdot 97 + \left(22 a + 30\right)\cdot 97^{2} + \left(43 a + 73\right)\cdot 97^{3} + \left(54 a + 85\right)\cdot 97^{4} + \left(35 a + 23\right)\cdot 97^{5} + \left(91 a + 30\right)\cdot 97^{6} + \left(47 a + 34\right)\cdot 97^{7} + \left(33 a + 2\right)\cdot 97^{8} + 75 a\cdot 97^{9} +O(97^{10})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 27 a + 92 + \left(93 a + 33\right)\cdot 97 + \left(9 a + 7\right)\cdot 97^{2} + \left(35 a + 38\right)\cdot 97^{3} + \left(31 a + 17\right)\cdot 97^{4} + \left(25 a + 79\right)\cdot 97^{5} + \left(83 a + 83\right)\cdot 97^{6} + \left(50 a + 24\right)\cdot 97^{7} + \left(12 a + 3\right)\cdot 97^{8} + \left(82 a + 20\right)\cdot 97^{9} +O(97^{10})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 12 a + 42 + \left(41 a + 55\right)\cdot 97 + \left(51 a + 56\right)\cdot 97^{2} + \left(15 a + 44\right)\cdot 97^{3} + \left(86 a + 79\right)\cdot 97^{4} + \left(9 a + 95\right)\cdot 97^{5} + \left(90 a + 60\right)\cdot 97^{6} + \left(51 a + 34\right)\cdot 97^{7} + \left(63 a + 35\right)\cdot 97^{8} + \left(16 a + 1\right)\cdot 97^{9} +O(97^{10})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 93 a + 10 + \left(55 a + 47\right)\cdot 97 + \left(73 a + 91\right)\cdot 97^{2} + \left(17 a + 60\right)\cdot 97^{3} + \left(73 a + 41\right)\cdot 97^{4} + \left(90 a + 85\right)\cdot 97^{5} + \left(4 a + 61\right)\cdot 97^{6} + \left(31 a + 95\right)\cdot 97^{7} + \left(41 a + 22\right)\cdot 97^{8} + \left(64 a + 80\right)\cdot 97^{9} +O(97^{10})\)  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.