Properties

Label 18.148...064.36t1758.a.a
Dimension $18$
Group $S_4\wr C_2$
Conductor $1.485\times 10^{30}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $18$
Group: $S_4\wr C_2$
Conductor: \(148\!\cdots\!064\)\(\medspace = 2^{40} \cdot 3^{38}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 8.2.11609505792.9
Galois orbit size: $1$
Smallest permutation container: 36T1758
Parity: odd
Determinant: 1.4.2t1.a.a
Projective image: $S_4\wr C_2$
Projective stem field: Galois closure of 8.2.11609505792.9

Defining polynomial

$f(x)$$=$ \( x^{8} - 4x^{7} + 4x^{6} + 21x^{4} - 84x^{3} + 132x^{2} - 96x + 24 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 131 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 131 }$: \( x^{2} + 127x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 18 a + 113 + \left(108 a + 120\right)\cdot 131 + \left(73 a + 115\right)\cdot 131^{2} + \left(3 a + 99\right)\cdot 131^{3} + \left(a + 50\right)\cdot 131^{4} + \left(19 a + 63\right)\cdot 131^{5} + \left(30 a + 55\right)\cdot 131^{6} + \left(7 a + 31\right)\cdot 131^{7} + \left(52 a + 2\right)\cdot 131^{8} + \left(28 a + 67\right)\cdot 131^{9} +O(131^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 26 + 96\cdot 131 + 62\cdot 131^{2} + 6\cdot 131^{3} + 71\cdot 131^{4} + 26\cdot 131^{5} + 96\cdot 131^{6} + 97\cdot 131^{7} + 100\cdot 131^{8} + 27\cdot 131^{9} +O(131^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 20 + 84\cdot 131 + 10\cdot 131^{2} + 83\cdot 131^{3} + 56\cdot 131^{4} + 77\cdot 131^{5} + 131^{6} + 54\cdot 131^{7} + 75\cdot 131^{8} + 40\cdot 131^{9} +O(131^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 128 a + 77 + \left(69 a + 121\right)\cdot 131 + \left(46 a + 8\right)\cdot 131^{2} + \left(73 a + 56\right)\cdot 131^{3} + \left(129 a + 115\right)\cdot 131^{4} + \left(31 a + 66\right)\cdot 131^{5} + \left(27 a + 43\right)\cdot 131^{6} + \left(71 a + 61\right)\cdot 131^{7} + \left(18 a + 130\right)\cdot 131^{8} + \left(70 a + 44\right)\cdot 131^{9} +O(131^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 113 a + 54 + \left(22 a + 11\right)\cdot 131 + \left(57 a + 41\right)\cdot 131^{2} + \left(127 a + 40\right)\cdot 131^{3} + \left(129 a + 51\right)\cdot 131^{4} + \left(111 a + 7\right)\cdot 131^{5} + \left(100 a + 26\right)\cdot 131^{6} + \left(123 a + 30\right)\cdot 131^{7} + \left(78 a + 72\right)\cdot 131^{8} + \left(102 a + 128\right)\cdot 131^{9} +O(131^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 3 a + 65 + \left(61 a + 11\right)\cdot 131 + \left(84 a + 125\right)\cdot 131^{2} + \left(57 a + 40\right)\cdot 131^{3} + \left(a + 36\right)\cdot 131^{4} + \left(99 a + 65\right)\cdot 131^{5} + \left(103 a + 120\right)\cdot 131^{6} + \left(59 a + 56\right)\cdot 131^{7} + \left(112 a + 2\right)\cdot 131^{8} + \left(60 a + 45\right)\cdot 131^{9} +O(131^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 54 a + 44 + \left(46 a + 104\right)\cdot 131 + \left(44 a + 79\right)\cdot 131^{2} + \left(115 a + 86\right)\cdot 131^{3} + \left(26 a + 9\right)\cdot 131^{4} + \left(14 a + 28\right)\cdot 131^{5} + \left(6 a + 85\right)\cdot 131^{6} + \left(117 a + 61\right)\cdot 131^{7} + \left(33 a + 126\right)\cdot 131^{8} + \left(28 a + 110\right)\cdot 131^{9} +O(131^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 77 a + 129 + \left(84 a + 104\right)\cdot 131 + \left(86 a + 79\right)\cdot 131^{2} + \left(15 a + 110\right)\cdot 131^{3} + \left(104 a + 1\right)\cdot 131^{4} + \left(116 a + 58\right)\cdot 131^{5} + \left(124 a + 95\right)\cdot 131^{6} + \left(13 a + 130\right)\cdot 131^{7} + \left(97 a + 13\right)\cdot 131^{8} + \left(102 a + 59\right)\cdot 131^{9} +O(131^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.