Properties

Label 45.163...288.336.a.b
Dimension $45$
Group $A_8$
Conductor $1.639\times 10^{319}$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $45$
Group: $A_8$
Conductor: \(163\!\cdots\!288\)\(\medspace = 2^{138} \cdot 11^{39} \cdot 151^{39} \cdot 7933^{39}\)
Artin stem field: Galois closure of 8.0.87813728302462049260764173611685476867240408121344.1
Galois orbit size: $2$
Smallest permutation container: 336
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $A_8$
Projective stem field: Galois closure of 8.0.87813728302462049260764173611685476867240408121344.1

Defining polynomial

$f(x)$$=$ \( x^{8} - 112x^{6} - 896x^{5} - 3360x^{4} - 7168x^{3} - 8960x^{2} - 6144x + 210825616 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 157 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 157 }$: \( x^{2} + 152x + 5 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 22 + 125\cdot 157 + 132\cdot 157^{2} + 119\cdot 157^{3} + 141\cdot 157^{4} + 34\cdot 157^{5} + 19\cdot 157^{6} + 121\cdot 157^{7} + 141\cdot 157^{8} + 129\cdot 157^{9} +O(157^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 106 a + 115 + \left(77 a + 101\right)\cdot 157 + \left(13 a + 85\right)\cdot 157^{2} + \left(38 a + 155\right)\cdot 157^{3} + \left(127 a + 126\right)\cdot 157^{4} + \left(29 a + 88\right)\cdot 157^{5} + \left(52 a + 121\right)\cdot 157^{6} + \left(104 a + 114\right)\cdot 157^{7} + \left(59 a + 79\right)\cdot 157^{8} + \left(40 a + 67\right)\cdot 157^{9} +O(157^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 103 a + 100 + \left(155 a + 22\right)\cdot 157 + \left(87 a + 51\right)\cdot 157^{2} + \left(112 a + 106\right)\cdot 157^{3} + \left(74 a + 121\right)\cdot 157^{4} + \left(27 a + 93\right)\cdot 157^{5} + \left(98 a + 80\right)\cdot 157^{6} + \left(60 a + 51\right)\cdot 157^{7} + \left(125 a + 7\right)\cdot 157^{8} + \left(6 a + 144\right)\cdot 157^{9} +O(157^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 124 + 154\cdot 157 + 97\cdot 157^{2} + 27\cdot 157^{3} + 5\cdot 157^{4} + 157^{5} + 22\cdot 157^{6} + 64\cdot 157^{7} + 128\cdot 157^{8} + 148\cdot 157^{9} +O(157^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 117 + 111\cdot 157 + 145\cdot 157^{2} + 32\cdot 157^{3} + 49\cdot 157^{4} + 27\cdot 157^{5} + 121\cdot 157^{6} + 32\cdot 157^{7} + 34\cdot 157^{8} + 155\cdot 157^{9} +O(157^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 54 a + 144 + \left(a + 69\right)\cdot 157 + \left(69 a + 21\right)\cdot 157^{2} + \left(44 a + 110\right)\cdot 157^{3} + \left(82 a + 68\right)\cdot 157^{4} + \left(129 a + 156\right)\cdot 157^{5} + \left(58 a + 72\right)\cdot 157^{6} + \left(96 a + 99\right)\cdot 157^{7} + \left(31 a + 102\right)\cdot 157^{8} + \left(150 a + 52\right)\cdot 157^{9} +O(157^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 146 + 128\cdot 157 + 17\cdot 157^{2} + 57\cdot 157^{3} + 17\cdot 157^{4} + 115\cdot 157^{5} + 151\cdot 157^{6} + 30\cdot 157^{7} + 17\cdot 157^{8} + 34\cdot 157^{9} +O(157^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 51 a + 17 + \left(79 a + 70\right)\cdot 157 + \left(143 a + 75\right)\cdot 157^{2} + \left(118 a + 18\right)\cdot 157^{3} + \left(29 a + 97\right)\cdot 157^{4} + \left(127 a + 110\right)\cdot 157^{5} + \left(104 a + 38\right)\cdot 157^{6} + \left(52 a + 113\right)\cdot 157^{7} + \left(97 a + 116\right)\cdot 157^{8} + \left(116 a + 52\right)\cdot 157^{9} +O(157^{10})\) Copy content 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$$1$$()$$45$
$105$$2$$(1,2)(3,4)(5,6)(7,8)$$-3$
$210$$2$$(1,2)(3,4)$$-3$
$112$$3$$(1,2,3)$$0$
$1120$$3$$(1,2,3)(4,5,6)$$0$
$1260$$4$$(1,2,3,4)(5,6,7,8)$$1$
$2520$$4$$(1,2,3,4)(5,6)$$1$
$1344$$5$$(1,2,3,4,5)$$0$
$1680$$6$$(1,2,3)(4,5)(6,7)$$0$
$3360$$6$$(1,2,3,4,5,6)(7,8)$$0$
$2880$$7$$(1,2,3,4,5,6,7)$$-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$
$2880$$7$$(1,3,4,5,6,7,2)$$\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$
$1344$$15$$(1,2,3,4,5)(6,7,8)$$0$
$1344$$15$$(1,3,4,5,2)(6,7,8)$$0$

The blue line marks the conjugacy class containing complex conjugation.