Properties

Label 18.831...000.36t1758.a.a
Dimension $18$
Group $S_4\wr C_2$
Conductor $8.310\times 10^{30}$
Root number $1$
Indicator $1$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $18$
Group: $S_4\wr C_2$
Conductor: \(831\!\cdots\!000\)\(\medspace = 2^{59} \cdot 3^{10} \cdot 5^{12}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 8.2.5898240000.1
Galois orbit size: $1$
Smallest permutation container: 36T1758
Parity: odd
Determinant: 1.8.2t1.b.a
Projective image: $S_4\wr C_2$
Projective stem field: Galois closure of 8.2.5898240000.1

Defining polynomial

$f(x)$$=$ \( x^{8} - 2x^{6} + x^{4} - 8x^{3} - 14x^{2} - 8x - 1 \) 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:
$r_{ 1 }$ $=$ \( 124 a + 90 + \left(160 a + 147\right)\cdot 167 + \left(116 a + 88\right)\cdot 167^{2} + \left(113 a + 27\right)\cdot 167^{3} + \left(163 a + 102\right)\cdot 167^{4} + \left(120 a + 27\right)\cdot 167^{5} + \left(162 a + 10\right)\cdot 167^{6} + \left(164 a + 28\right)\cdot 167^{7} + \left(2 a + 15\right)\cdot 167^{8} + \left(51 a + 24\right)\cdot 167^{9} +O(167^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 32 + 63\cdot 167 + 161\cdot 167^{2} + 19\cdot 167^{3} + 127\cdot 167^{4} + 106\cdot 167^{5} + 141\cdot 167^{6} + 120\cdot 167^{7} + 41\cdot 167^{8} + 47\cdot 167^{9} +O(167^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 43 a + 68 + \left(38 a + 117\right)\cdot 167 + \left(5 a + 142\right)\cdot 167^{2} + \left(96 a + 140\right)\cdot 167^{3} + \left(114 a + 160\right)\cdot 167^{4} + \left(89 a + 130\right)\cdot 167^{5} + \left(48 a + 106\right)\cdot 167^{6} + \left(85 a + 95\right)\cdot 167^{7} + \left(133 a + 63\right)\cdot 167^{8} + \left(37 a + 155\right)\cdot 167^{9} +O(167^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 43 a + 47 + \left(6 a + 17\right)\cdot 167 + \left(50 a + 45\right)\cdot 167^{2} + \left(53 a + 24\right)\cdot 167^{3} + \left(3 a + 152\right)\cdot 167^{4} + \left(46 a + 151\right)\cdot 167^{5} + \left(4 a + 51\right)\cdot 167^{6} + \left(2 a + 30\right)\cdot 167^{7} + \left(164 a + 20\right)\cdot 167^{8} + \left(115 a + 72\right)\cdot 167^{9} +O(167^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 124 a + 111 + \left(128 a + 112\right)\cdot 167 + \left(161 a + 109\right)\cdot 167^{2} + \left(70 a + 64\right)\cdot 167^{3} + \left(52 a + 12\right)\cdot 167^{4} + \left(77 a + 106\right)\cdot 167^{5} + \left(118 a + 65\right)\cdot 167^{6} + \left(81 a + 132\right)\cdot 167^{7} + \left(33 a + 111\right)\cdot 167^{8} + \left(129 a + 59\right)\cdot 167^{9} +O(167^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 165 + 105\cdot 167 + 38\cdot 167^{2} + 95\cdot 167^{3} + 119\cdot 167^{4} + 47\cdot 167^{5} + 130\cdot 167^{6} + 154\cdot 167^{7} + 89\cdot 167^{8} + 23\cdot 167^{9} +O(167^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 57 a + 49 + \left(110 a + 25\right)\cdot 167 + \left(61 a + 65\right)\cdot 167^{2} + \left(39 a + 75\right)\cdot 167^{3} + \left(4 a + 14\right)\cdot 167^{4} + \left(118 a + 75\right)\cdot 167^{5} + \left(99 a + 6\right)\cdot 167^{6} + \left(21 a + 92\right)\cdot 167^{7} + \left(11 a + 84\right)\cdot 167^{8} + \left(162 a + 67\right)\cdot 167^{9} +O(167^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 110 a + 106 + \left(56 a + 78\right)\cdot 167 + \left(105 a + 16\right)\cdot 167^{2} + \left(127 a + 53\right)\cdot 167^{3} + \left(162 a + 146\right)\cdot 167^{4} + \left(48 a + 21\right)\cdot 167^{5} + \left(67 a + 155\right)\cdot 167^{6} + \left(145 a + 13\right)\cdot 167^{7} + \left(155 a + 74\right)\cdot 167^{8} + \left(4 a + 51\right)\cdot 167^{9} +O(167^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$18$
$6$$2$$(1,4)(2,6)$$-6$
$9$$2$$(1,4)(2,6)(3,7)(5,8)$$2$
$12$$2$$(3,5)$$0$
$24$$2$$(1,3)(2,5)(4,7)(6,8)$$0$
$36$$2$$(1,2)(3,5)$$-2$
$36$$2$$(1,4)(2,6)(3,5)$$0$
$16$$3$$(3,7,8)$$0$
$64$$3$$(2,4,6)(3,7,8)$$0$
$12$$4$$(1,2,4,6)$$0$
$36$$4$$(1,2,4,6)(3,5,7,8)$$-2$
$36$$4$$(1,4)(2,6)(3,5,7,8)$$0$
$72$$4$$(1,7,4,3)(2,8,6,5)$$0$
$72$$4$$(1,2,4,6)(3,5)$$2$
$144$$4$$(1,3,2,5)(4,7)(6,8)$$0$
$48$$6$$(1,4)(2,6)(3,8,7)$$0$
$96$$6$$(2,6,4)(3,5)$$0$
$192$$6$$(1,5)(2,7,4,8,6,3)$$0$
$144$$8$$(1,5,2,7,4,8,6,3)$$0$
$96$$12$$(1,2,4,6)(3,7,8)$$0$

The blue line marks the conjugacy class containing complex conjugation.