Properties

Label 9.318916345856.16t1294.a.a
Dimension $9$
Group $S_4\wr C_2$
Conductor $318916345856$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $9$
Group: $S_4\wr C_2$
Conductor: \(318916345856\)\(\medspace = 2^{14} \cdot 269^{3} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 8.2.335111316544.1
Galois orbit size: $1$
Smallest permutation container: 16T1294
Parity: odd
Determinant: 1.1076.2t1.a.a
Projective image: $S_4\wr C_2$
Projective stem field: Galois closure of 8.2.335111316544.1

Defining polynomial

$f(x)$$=$ \( x^{8} - 2x^{7} - 10x^{6} + 7x^{5} + 5x^{4} - 16x^{3} + 64x^{2} - 76x + 92 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 191 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 191 }$: \( x^{2} + 190x + 19 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 75 + 134\cdot 191 + 50\cdot 191^{2} + 140\cdot 191^{3} + 87\cdot 191^{4} + 97\cdot 191^{5} + 28\cdot 191^{6} + 67\cdot 191^{7} + 6\cdot 191^{8} + 70\cdot 191^{9} +O(191^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 35 a + 95 + \left(15 a + 32\right)\cdot 191 + \left(14 a + 13\right)\cdot 191^{2} + \left(36 a + 39\right)\cdot 191^{3} + \left(98 a + 162\right)\cdot 191^{4} + \left(117 a + 129\right)\cdot 191^{5} + \left(71 a + 176\right)\cdot 191^{6} + \left(156 a + 28\right)\cdot 191^{7} + \left(133 a + 67\right)\cdot 191^{8} + \left(189 a + 188\right)\cdot 191^{9} +O(191^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 60 a + 98 + \left(28 a + 29\right)\cdot 191 + \left(49 a + 8\right)\cdot 191^{2} + \left(84 a + 19\right)\cdot 191^{3} + \left(51 a + 124\right)\cdot 191^{4} + \left(109 a + 44\right)\cdot 191^{5} + \left(175 a + 94\right)\cdot 191^{6} + \left(90 a + 137\right)\cdot 191^{7} + \left(173 a + 40\right)\cdot 191^{8} + \left(104 a + 139\right)\cdot 191^{9} +O(191^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 116 a + 101 + \left(104 a + 87\right)\cdot 191 + \left(23 a + 117\right)\cdot 191^{2} + \left(a + 165\right)\cdot 191^{3} + \left(25 a + 166\right)\cdot 191^{4} + \left(59 a + 4\right)\cdot 191^{5} + \left(128 a + 29\right)\cdot 191^{6} + \left(95 a + 112\right)\cdot 191^{7} + \left(67 a + 27\right)\cdot 191^{8} + \left(190 a + 120\right)\cdot 191^{9} +O(191^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 156 a + 130 + \left(175 a + 12\right)\cdot 191 + \left(176 a + 12\right)\cdot 191^{2} + \left(154 a + 61\right)\cdot 191^{3} + \left(92 a + 33\right)\cdot 191^{4} + \left(73 a + 149\right)\cdot 191^{5} + \left(119 a + 130\right)\cdot 191^{6} + \left(34 a + 113\right)\cdot 191^{7} + \left(57 a + 44\right)\cdot 191^{8} + \left(a + 53\right)\cdot 191^{9} +O(191^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 83 + 11\cdot 191 + 115\cdot 191^{2} + 141\cdot 191^{3} + 98\cdot 191^{4} + 5\cdot 191^{5} + 46\cdot 191^{6} + 172\cdot 191^{7} + 72\cdot 191^{8} + 70\cdot 191^{9} +O(191^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 131 a + 158 + \left(162 a + 188\right)\cdot 191 + \left(141 a + 28\right)\cdot 191^{2} + \left(106 a + 54\right)\cdot 191^{3} + \left(139 a + 91\right)\cdot 191^{4} + \left(81 a + 102\right)\cdot 191^{5} + \left(15 a + 160\right)\cdot 191^{6} + \left(100 a + 52\right)\cdot 191^{7} + \left(17 a + 123\right)\cdot 191^{8} + \left(86 a + 70\right)\cdot 191^{9} +O(191^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 75 a + 26 + \left(86 a + 76\right)\cdot 191 + \left(167 a + 36\right)\cdot 191^{2} + \left(189 a + 143\right)\cdot 191^{3} + \left(165 a + 190\right)\cdot 191^{4} + \left(131 a + 38\right)\cdot 191^{5} + \left(62 a + 98\right)\cdot 191^{6} + \left(95 a + 79\right)\cdot 191^{7} + \left(123 a + 190\right)\cdot 191^{8} + 51\cdot 191^{9} +O(191^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.