Properties

Label 8.713...025.24t708.d.a
Dimension $8$
Group $C_2 \wr S_4$
Conductor $7.139\times 10^{12}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $8$
Group: $C_2 \wr S_4$
Conductor: \(7138541958025\)\(\medspace = 5^{2} \cdot 17^{4} \cdot 43^{4} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 8.0.2671805.1
Galois orbit size: $1$
Smallest permutation container: 24T708
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $C_2^3:S_4$
Projective stem field: Galois closure of 8.4.333975625.1

Defining polynomial

$f(x)$$=$ \( x^{8} - x^{5} + x^{4} - x^{3} + 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 167 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 167 }$: \( x^{2} + 166x + 5 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 6 a + 151 + \left(158 a + 132\right)\cdot 167 + \left(105 a + 38\right)\cdot 167^{2} + \left(19 a + 8\right)\cdot 167^{3} + \left(154 a + 117\right)\cdot 167^{4} + \left(133 a + 45\right)\cdot 167^{5} + \left(164 a + 38\right)\cdot 167^{6} + \left(99 a + 6\right)\cdot 167^{7} + \left(36 a + 144\right)\cdot 167^{8} + \left(8 a + 74\right)\cdot 167^{9} +O(167^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 105 + 49\cdot 167 + 140\cdot 167^{2} + 35\cdot 167^{3} + 154\cdot 167^{4} + 110\cdot 167^{5} + 97\cdot 167^{6} + 160\cdot 167^{7} + 15\cdot 167^{8} + 89\cdot 167^{9} +O(167^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 63 a + 25 + \left(102 a + 125\right)\cdot 167 + \left(33 a + 7\right)\cdot 167^{2} + \left(37 a + 147\right)\cdot 167^{3} + \left(56 a + 43\right)\cdot 167^{4} + \left(78 a + 104\right)\cdot 167^{5} + \left(145 a + 96\right)\cdot 167^{6} + \left(72 a + 100\right)\cdot 167^{7} + \left(97 a + 151\right)\cdot 167^{8} + \left(165 a + 143\right)\cdot 167^{9} +O(167^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( a + 53 + \left(40 a + 106\right)\cdot 167 + \left(135 a + 89\right)\cdot 167^{2} + \left(154 a + 9\right)\cdot 167^{3} + \left(84 a + 61\right)\cdot 167^{4} + \left(147 a + 147\right)\cdot 167^{5} + \left(153 a + 137\right)\cdot 167^{6} + \left(93 a + 143\right)\cdot 167^{7} + \left(155 a + 9\right)\cdot 167^{8} + \left(81 a + 3\right)\cdot 167^{9} +O(167^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 161 a + 157 + \left(8 a + 117\right)\cdot 167 + \left(61 a + 153\right)\cdot 167^{2} + \left(147 a + 88\right)\cdot 167^{3} + \left(12 a + 84\right)\cdot 167^{4} + \left(33 a + 25\right)\cdot 167^{5} + \left(2 a + 69\right)\cdot 167^{6} + \left(67 a + 108\right)\cdot 167^{7} + \left(130 a + 80\right)\cdot 167^{8} + \left(158 a + 46\right)\cdot 167^{9} +O(167^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 104 a + 88 + \left(64 a + 164\right)\cdot 167 + \left(133 a + 105\right)\cdot 167^{2} + \left(129 a + 150\right)\cdot 167^{3} + \left(110 a + 62\right)\cdot 167^{4} + \left(88 a + 126\right)\cdot 167^{5} + \left(21 a + 163\right)\cdot 167^{6} + \left(94 a + 27\right)\cdot 167^{7} + \left(69 a + 9\right)\cdot 167^{8} + \left(a + 45\right)\cdot 167^{9} +O(167^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 35 + 160\cdot 167 + 113\cdot 167^{2} + 31\cdot 167^{3} + 153\cdot 167^{4} + 64\cdot 167^{5} + 87\cdot 167^{6} + 36\cdot 167^{7} + 18\cdot 167^{8} + 2\cdot 167^{9} +O(167^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 166 a + 54 + \left(126 a + 145\right)\cdot 167 + \left(31 a + 17\right)\cdot 167^{2} + \left(12 a + 29\right)\cdot 167^{3} + \left(82 a + 158\right)\cdot 167^{4} + \left(19 a + 42\right)\cdot 167^{5} + \left(13 a + 144\right)\cdot 167^{6} + \left(73 a + 83\right)\cdot 167^{7} + \left(11 a + 71\right)\cdot 167^{8} + \left(85 a + 96\right)\cdot 167^{9} +O(167^{10})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

Cycle notation
$(1,2,3,4)(5,8,7,6)$
$(1,2)(7,8)$
$(1,8)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$8$
$1$$2$$(1,8)(2,7)(3,6)(4,5)$$-8$
$4$$2$$(3,6)$$4$
$4$$2$$(2,7)(3,6)(4,5)$$-4$
$6$$2$$(1,8)(3,6)$$0$
$12$$2$$(1,3)(2,4)(5,7)(6,8)$$0$
$12$$2$$(1,2)(7,8)$$0$
$12$$2$$(1,4)(2,7)(3,6)(5,8)$$0$
$24$$2$$(1,2)(3,6)(7,8)$$0$
$32$$3$$(1,3,4)(5,8,6)$$-1$
$12$$4$$(1,3,8,6)(2,4,7,5)$$0$
$12$$4$$(1,2,8,7)$$0$
$12$$4$$(1,8)(2,7)(3,5,6,4)$$0$
$24$$4$$(1,3,8,6)(2,4)(5,7)$$0$
$24$$4$$(1,2,8,7)(3,6)$$0$
$48$$4$$(1,2,3,4)(5,8,7,6)$$0$
$32$$6$$(2,4,3,7,5,6)$$-1$
$32$$6$$(1,3,4)(2,7)(5,8,6)$$1$
$32$$6$$(1,3,5,8,6,4)(2,7)$$1$
$48$$8$$(1,4,3,7,8,5,6,2)$$0$

The blue line marks the conjugacy class containing complex conjugation.