Properties

Label 45.135...232.336.a.b
Dimension $45$
Group $A_8$
Conductor $1.356\times 10^{295}$
Root number not computed
Indicator $0$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $45$
Group: $A_8$
Conductor: \(135\!\cdots\!232\)\(\medspace = 2^{136} \cdot 3294173^{39}\)
Artin stem field: Galois closure of 8.0.1339918344154038594028110691225010840890507264.1
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 8.0.1339918344154038594028110691225010840890507264.1

Defining polynomial

$f(x)$$=$ \( x^{8} - 112x^{6} - 896x^{5} - 3360x^{4} - 7168x^{3} - 8960x^{2} - 6144x + 210825280 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 197 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 197 }$: \( x^{2} + 192x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 59 a + 194 + \left(126 a + 186\right)\cdot 197 + \left(165 a + 86\right)\cdot 197^{2} + \left(51 a + 70\right)\cdot 197^{3} + \left(47 a + 171\right)\cdot 197^{4} + \left(65 a + 39\right)\cdot 197^{5} + \left(146 a + 134\right)\cdot 197^{6} + \left(25 a + 2\right)\cdot 197^{7} + 6 a\cdot 197^{8} + \left(35 a + 55\right)\cdot 197^{9} +O(197^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 138 a + 95 + \left(70 a + 168\right)\cdot 197 + 31 a\cdot 197^{2} + \left(145 a + 164\right)\cdot 197^{3} + \left(149 a + 158\right)\cdot 197^{4} + \left(131 a + 121\right)\cdot 197^{5} + \left(50 a + 12\right)\cdot 197^{6} + \left(171 a + 182\right)\cdot 197^{7} + \left(190 a + 4\right)\cdot 197^{8} + \left(161 a + 27\right)\cdot 197^{9} +O(197^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 111 a + 52 + \left(191 a + 109\right)\cdot 197 + \left(43 a + 4\right)\cdot 197^{2} + \left(155 a + 163\right)\cdot 197^{3} + \left(64 a + 78\right)\cdot 197^{4} + \left(117 a + 32\right)\cdot 197^{5} + \left(169 a + 146\right)\cdot 197^{6} + \left(96 a + 69\right)\cdot 197^{7} + \left(3 a + 148\right)\cdot 197^{8} + 52 a\cdot 197^{9} +O(197^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 86 a + 16 + \left(5 a + 168\right)\cdot 197 + \left(153 a + 32\right)\cdot 197^{2} + \left(41 a + 107\right)\cdot 197^{3} + \left(132 a + 50\right)\cdot 197^{4} + \left(79 a + 160\right)\cdot 197^{5} + \left(27 a + 88\right)\cdot 197^{6} + \left(100 a + 187\right)\cdot 197^{7} + \left(193 a + 68\right)\cdot 197^{8} + \left(144 a + 60\right)\cdot 197^{9} +O(197^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 76 a + 83 + \left(155 a + 143\right)\cdot 197 + \left(196 a + 151\right)\cdot 197^{2} + \left(184 a + 144\right)\cdot 197^{3} + \left(72 a + 167\right)\cdot 197^{4} + \left(70 a + 43\right)\cdot 197^{5} + \left(103 a + 157\right)\cdot 197^{6} + \left(42 a + 150\right)\cdot 197^{7} + \left(178 a + 184\right)\cdot 197^{8} + \left(17 a + 62\right)\cdot 197^{9} +O(197^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 32 a + 158 + \left(53 a + 57\right)\cdot 197 + \left(77 a + 91\right)\cdot 197^{2} + \left(33 a + 80\right)\cdot 197^{3} + \left(74 a + 33\right)\cdot 197^{4} + \left(155 a + 76\right)\cdot 197^{5} + \left(183 a + 130\right)\cdot 197^{6} + \left(158 a + 154\right)\cdot 197^{7} + \left(7 a + 127\right)\cdot 197^{8} + \left(194 a + 19\right)\cdot 197^{9} +O(197^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 165 a + 121 + \left(143 a + 94\right)\cdot 197 + \left(119 a + 30\right)\cdot 197^{2} + \left(163 a + 170\right)\cdot 197^{3} + \left(122 a + 173\right)\cdot 197^{4} + \left(41 a + 187\right)\cdot 197^{5} + \left(13 a + 105\right)\cdot 197^{6} + \left(38 a + 174\right)\cdot 197^{7} + \left(189 a + 7\right)\cdot 197^{8} + \left(2 a + 194\right)\cdot 197^{9} +O(197^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 121 a + 69 + \left(41 a + 56\right)\cdot 197 + 192\cdot 197^{2} + \left(12 a + 84\right)\cdot 197^{3} + \left(124 a + 150\right)\cdot 197^{4} + \left(126 a + 125\right)\cdot 197^{5} + \left(93 a + 12\right)\cdot 197^{6} + \left(154 a + 63\right)\cdot 197^{7} + \left(18 a + 48\right)\cdot 197^{8} + \left(179 a + 171\right)\cdot 197^{9} +O(197^{10})\) 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.