Properties

Label 45.761...136.336.a.a
Dimension $45$
Group $A_8$
Conductor $7.610\times 10^{229}$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $45$
Group: $A_8$
Conductor: \(761\!\cdots\!136\)\(\medspace = 2^{176} \cdot 7^{56} \cdot 11^{39} \cdot 191^{39}\)
Artin stem field: Galois closure of 8.0.133100753213221593424899389161209856.1
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 8.0.133100753213221593424899389161209856.1

Defining polynomial

$f(x)$$=$ \( x^{8} - 28x^{6} - 112x^{5} - 210x^{4} - 224x^{3} - 140x^{2} - 48x + 823585 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 193 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 193 }$: \( x^{2} + 192x + 5 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 121 a + 67 + \left(178 a + 135\right)\cdot 193 + \left(181 a + 110\right)\cdot 193^{2} + \left(150 a + 160\right)\cdot 193^{3} + \left(66 a + 82\right)\cdot 193^{4} + \left(138 a + 189\right)\cdot 193^{5} + \left(151 a + 164\right)\cdot 193^{6} + \left(46 a + 192\right)\cdot 193^{7} + \left(114 a + 92\right)\cdot 193^{8} + \left(140 a + 174\right)\cdot 193^{9} +O(193^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 76 + 117\cdot 193 + 102\cdot 193^{2} + 88\cdot 193^{3} + 74\cdot 193^{4} + 103\cdot 193^{5} + 84\cdot 193^{6} + 153\cdot 193^{7} + 141\cdot 193^{8} + 91\cdot 193^{9} +O(193^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 11 a + 191 + \left(162 a + 14\right)\cdot 193 + \left(68 a + 131\right)\cdot 193^{2} + \left(114 a + 104\right)\cdot 193^{3} + \left(154 a + 71\right)\cdot 193^{4} + \left(133 a + 166\right)\cdot 193^{5} + \left(132 a + 13\right)\cdot 193^{6} + \left(173 a + 85\right)\cdot 193^{7} + \left(21 a + 87\right)\cdot 193^{8} + \left(98 a + 104\right)\cdot 193^{9} +O(193^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 114 + 166\cdot 193 + 29\cdot 193^{2} + 89\cdot 193^{3} + 134\cdot 193^{4} + 74\cdot 193^{5} + 150\cdot 193^{6} + 70\cdot 193^{7} + 65\cdot 193^{8} + 5\cdot 193^{9} +O(193^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 182 a + 9 + \left(30 a + 166\right)\cdot 193 + \left(124 a + 37\right)\cdot 193^{2} + \left(78 a + 150\right)\cdot 193^{3} + \left(38 a + 111\right)\cdot 193^{4} + \left(59 a + 145\right)\cdot 193^{5} + \left(60 a + 12\right)\cdot 193^{6} + \left(19 a + 126\right)\cdot 193^{7} + \left(171 a + 128\right)\cdot 193^{8} + \left(94 a + 180\right)\cdot 193^{9} +O(193^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 95 + 15\cdot 193 + 126\cdot 193^{2} + 150\cdot 193^{3} + 70\cdot 193^{4} + 162\cdot 193^{5} + 41\cdot 193^{6} + 48\cdot 193^{7} + 121\cdot 193^{8} + 17\cdot 193^{9} +O(193^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 72 a + 188 + \left(14 a + 192\right)\cdot 193 + \left(11 a + 113\right)\cdot 193^{2} + \left(42 a + 129\right)\cdot 193^{3} + \left(126 a + 191\right)\cdot 193^{4} + \left(54 a + 67\right)\cdot 193^{5} + \left(41 a + 178\right)\cdot 193^{6} + \left(146 a + 87\right)\cdot 193^{7} + \left(78 a + 160\right)\cdot 193^{8} + \left(52 a + 7\right)\cdot 193^{9} +O(193^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 32 + 156\cdot 193 + 119\cdot 193^{2} + 91\cdot 193^{3} + 34\cdot 193^{4} + 55\cdot 193^{5} + 125\cdot 193^{6} + 7\cdot 193^{7} + 167\cdot 193^{8} + 189\cdot 193^{9} +O(193^{10})\) 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)$$\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$
$2880$$7$$(1,3,4,5,6,7,2)$$-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$
$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.