Properties

Label 56.102...296.105.a.a
Dimension $56$
Group $A_8$
Conductor $1.026\times 10^{364}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $56$
Group: $A_8$
Conductor: \(102\!\cdots\!296\)\(\medspace = 2^{144} \cdot 11^{48} \cdot 435593^{48}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 8.0.3172352108695376607450650959096645338500694016.1
Galois orbit size: $1$
Smallest permutation container: 105
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $A_8$
Projective stem field: 8.0.3172352108695376607450650959096645338500694016.1

Defining polynomial

$f(x)$$=$\(x^{8} - 112 x^{6} - 896 x^{5} - 3360 x^{4} - 7168 x^{3} - 8960 x^{2} - 6144 x + 210825220\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$: \(x^{2} + 97 x + 2\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( a + 45 + \left(61 a + 21\right)\cdot 101 + \left(100 a + 77\right)\cdot 101^{2} + \left(57 a + 7\right)\cdot 101^{3} + \left(87 a + 82\right)\cdot 101^{4} + \left(92 a + 15\right)\cdot 101^{5} + \left(26 a + 55\right)\cdot 101^{6} + \left(58 a + 93\right)\cdot 101^{7} + \left(83 a + 93\right)\cdot 101^{8} + \left(65 a + 43\right)\cdot 101^{9} +O(101^{10})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 12 a + 27 + \left(32 a + 74\right)\cdot 101 + \left(59 a + 81\right)\cdot 101^{2} + \left(9 a + 36\right)\cdot 101^{3} + \left(75 a + 34\right)\cdot 101^{4} + \left(77 a + 51\right)\cdot 101^{5} + \left(38 a + 82\right)\cdot 101^{6} + \left(96 a + 27\right)\cdot 101^{7} + \left(73 a + 32\right)\cdot 101^{8} + \left(69 a + 74\right)\cdot 101^{9} +O(101^{10})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 74 a + 99 + \left(15 a + 1\right)\cdot 101 + \left(77 a + 10\right)\cdot 101^{2} + \left(99 a + 79\right)\cdot 101^{3} + \left(94 a + 98\right)\cdot 101^{4} + \left(59 a + 39\right)\cdot 101^{5} + \left(12 a + 42\right)\cdot 101^{6} + \left(77 a + 72\right)\cdot 101^{7} + \left(42 a + 12\right)\cdot 101^{8} + \left(a + 36\right)\cdot 101^{9} +O(101^{10})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 100 a + 49 + \left(39 a + 62\right)\cdot 101 + 14\cdot 101^{2} + \left(43 a + 38\right)\cdot 101^{3} + \left(13 a + 71\right)\cdot 101^{4} + \left(8 a + 97\right)\cdot 101^{5} + \left(74 a + 69\right)\cdot 101^{6} + \left(42 a + 97\right)\cdot 101^{7} + \left(17 a + 66\right)\cdot 101^{8} + \left(35 a + 21\right)\cdot 101^{9} +O(101^{10})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 89 a + 75 + \left(68 a + 89\right)\cdot 101 + \left(41 a + 84\right)\cdot 101^{2} + \left(91 a + 15\right)\cdot 101^{3} + \left(25 a + 22\right)\cdot 101^{4} + \left(23 a + 85\right)\cdot 101^{5} + \left(62 a + 58\right)\cdot 101^{6} + \left(4 a + 71\right)\cdot 101^{7} + \left(27 a + 29\right)\cdot 101^{8} + \left(31 a + 77\right)\cdot 101^{9} +O(101^{10})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 27 a + 92 + \left(85 a + 91\right)\cdot 101 + \left(23 a + 100\right)\cdot 101^{2} + \left(a + 97\right)\cdot 101^{3} + \left(6 a + 75\right)\cdot 101^{4} + \left(41 a + 83\right)\cdot 101^{5} + \left(88 a + 32\right)\cdot 101^{6} + \left(23 a + 65\right)\cdot 101^{7} + \left(58 a + 5\right)\cdot 101^{8} + \left(99 a + 100\right)\cdot 101^{9} +O(101^{10})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 65 a + 30 + \left(60 a + 43\right)\cdot 101 + \left(22 a + 2\right)\cdot 101^{2} + \left(29 a + 17\right)\cdot 101^{3} + \left(63 a + 49\right)\cdot 101^{4} + \left(10 a + 25\right)\cdot 101^{5} + \left(36 a + 65\right)\cdot 101^{6} + \left(2 a + 51\right)\cdot 101^{7} + \left(86 a + 11\right)\cdot 101^{8} + \left(53 a + 11\right)\cdot 101^{9} +O(101^{10})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 36 a + 88 + \left(40 a + 18\right)\cdot 101 + \left(78 a + 32\right)\cdot 101^{2} + \left(71 a + 10\right)\cdot 101^{3} + \left(37 a + 71\right)\cdot 101^{4} + \left(90 a + 4\right)\cdot 101^{5} + \left(64 a + 98\right)\cdot 101^{6} + \left(98 a + 24\right)\cdot 101^{7} + \left(14 a + 50\right)\cdot 101^{8} + \left(47 a + 39\right)\cdot 101^{9} +O(101^{10})\)  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.