Basic invariants
Dimension: | $12$ |
Group: | $S_3\wr S_3$ |
Conductor: | \(42865179747426752\)\(\medspace = 2^{6} \cdot 23^{5} \cdot 101^{4} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 9.1.1808892224.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 18T315 |
Parity: | odd |
Determinant: | 1.23.2t1.a.a |
Projective image: | $S_3\wr S_3$ |
Projective stem field: | Galois closure of 9.1.1808892224.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{9} - 4x^{7} + 4x^{5} - 12x^{4} - 12x^{3} + 8x^{2} - 8 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 131 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 131 }$: \( x^{3} + 3x + 129 \)
Roots:
$r_{ 1 }$ | $=$ | \( 28 a^{2} + 106 a + 27 + \left(16 a^{2} + 49 a + 87\right)\cdot 131 + \left(12 a^{2} + 110 a + 112\right)\cdot 131^{2} + \left(96 a^{2} + 60 a + 69\right)\cdot 131^{3} + \left(31 a^{2} + 3 a + 49\right)\cdot 131^{4} + \left(62 a^{2} + 60 a + 11\right)\cdot 131^{5} + \left(114 a^{2} + 101 a + 1\right)\cdot 131^{6} + \left(59 a^{2} + 105 a + 121\right)\cdot 131^{7} + \left(16 a^{2} + 78 a + 4\right)\cdot 131^{8} + \left(84 a^{2} + 19 a + 29\right)\cdot 131^{9} +O(131^{10})\) |
$r_{ 2 }$ | $=$ | \( 31 a^{2} + a + 116 + \left(113 a^{2} + 12 a + 88\right)\cdot 131 + \left(98 a^{2} + 83 a + 67\right)\cdot 131^{2} + \left(15 a^{2} + 38 a + 79\right)\cdot 131^{3} + \left(2 a^{2} + 27 a + 37\right)\cdot 131^{4} + \left(15 a^{2} + 27 a + 63\right)\cdot 131^{5} + \left(90 a^{2} + 55 a + 125\right)\cdot 131^{6} + \left(2 a^{2} + 35 a + 101\right)\cdot 131^{7} + \left(48 a^{2} + 60 a + 33\right)\cdot 131^{8} + \left(111 a^{2} + 96 a + 81\right)\cdot 131^{9} +O(131^{10})\) |
$r_{ 3 }$ | $=$ | \( 33 a^{2} + 64 a + 41 + \left(84 a^{2} + 4 a + 120\right)\cdot 131 + \left(65 a^{2} + 129 a + 41\right)\cdot 131^{2} + \left(a^{2} + 24 a + 77\right)\cdot 131^{3} + \left(48 a^{2} + 86 a + 76\right)\cdot 131^{4} + \left(92 a^{2} + 115 a + 2\right)\cdot 131^{5} + \left(95 a^{2} + 70 a + 81\right)\cdot 131^{6} + \left(48 a^{2} + 66 a + 130\right)\cdot 131^{7} + \left(a^{2} + 2 a + 92\right)\cdot 131^{8} + \left(45 a^{2} + 35 a + 108\right)\cdot 131^{9} +O(131^{10})\) |
$r_{ 4 }$ | $=$ | \( 36 a^{2} + 90 a + 43 + \left(50 a^{2} + 97 a + 24\right)\cdot 131 + \left(21 a^{2} + 101 a\right)\cdot 131^{2} + \left(52 a^{2} + 2 a + 113\right)\cdot 131^{3} + \left(18 a^{2} + 110 a + 22\right)\cdot 131^{4} + \left(45 a^{2} + 82 a + 108\right)\cdot 131^{5} + \left(71 a^{2} + 24 a + 45\right)\cdot 131^{6} + \left(122 a^{2} + 127 a + 115\right)\cdot 131^{7} + \left(32 a^{2} + 114 a + 37\right)\cdot 131^{8} + \left(72 a^{2} + 111 a + 5\right)\cdot 131^{9} +O(131^{10})\) |
$r_{ 5 }$ | $=$ | \( 39 a^{2} + 68 a + 1 + \left(74 a^{2} + 118 a + 11\right)\cdot 131 + \left(107 a^{2} + 101 a + 85\right)\cdot 131^{2} + \left(125 a^{2} + 32 a + 37\right)\cdot 131^{3} + \left(84 a^{2} + 108 a + 72\right)\cdot 131^{4} + \left(110 a + 34\right)\cdot 131^{5} + \left(68 a^{2} + 17 a + 81\right)\cdot 131^{6} + \left(34 a^{2} + 20 a + 34\right)\cdot 131^{7} + \left(94 a^{2} + 47 a + 126\right)\cdot 131^{8} + \left(83 a^{2} + 7 a + 25\right)\cdot 131^{9} +O(131^{10})\) |
$r_{ 6 }$ | $=$ | \( 42 a^{2} + 94 a + 59 + \left(40 a^{2} + 80 a + 32\right)\cdot 131 + \left(63 a^{2} + 74 a + 37\right)\cdot 131^{2} + \left(45 a^{2} + 10 a + 34\right)\cdot 131^{3} + \left(55 a^{2} + a + 91\right)\cdot 131^{4} + \left(84 a^{2} + 78 a + 117\right)\cdot 131^{5} + \left(43 a^{2} + 102 a + 107\right)\cdot 131^{6} + \left(108 a^{2} + 80 a + 118\right)\cdot 131^{7} + \left(125 a^{2} + 28 a + 79\right)\cdot 131^{8} + \left(110 a^{2} + 84 a + 109\right)\cdot 131^{9} +O(131^{10})\) |
$r_{ 7 }$ | $=$ | \( 56 a^{2} + 104 a + 87 + \left(6 a^{2} + 45 a + 95\right)\cdot 131 + \left(2 a^{2} + 58 a + 45\right)\cdot 131^{2} + \left(84 a^{2} + 95 a + 111\right)\cdot 131^{3} + \left(27 a^{2} + 43 a + 35\right)\cdot 131^{4} + \left(85 a^{2} + 68 a + 119\right)\cdot 131^{5} + \left(122 a^{2} + 88 a + 3\right)\cdot 131^{6} + \left(104 a^{2} + 114 a + 112\right)\cdot 131^{7} + \left(3 a^{2} + 99 a + 97\right)\cdot 131^{8} + \left(106 a^{2} + 11 a + 99\right)\cdot 131^{9} +O(131^{10})\) |
$r_{ 8 }$ | $=$ | \( 61 a^{2} + 62 a + 45 + \left(74 a^{2} + 11\right)\cdot 131 + \left(55 a^{2} + 77 a + 112\right)\cdot 131^{2} + \left(120 a^{2} + 59 a + 26\right)\cdot 131^{3} + \left(43 a^{2} + 126 a + 121\right)\cdot 131^{4} + \left(115 a^{2} + 123 a + 1\right)\cdot 131^{5} + \left(103 a^{2} + 57 a + 22\right)\cdot 131^{6} + \left(93 a^{2} + 75 a + 22\right)\cdot 131^{7} + \left(119 a^{2} + 23 a + 46\right)\cdot 131^{8} + \left(66 a^{2} + 27 a + 123\right)\cdot 131^{9} +O(131^{10})\) |
$r_{ 9 }$ | $=$ | \( 67 a^{2} + 66 a + 105 + \left(64 a^{2} + 114 a + 52\right)\cdot 131 + \left(97 a^{2} + 49 a + 21\right)\cdot 131^{2} + \left(113 a^{2} + 67 a + 105\right)\cdot 131^{3} + \left(80 a^{2} + 17 a + 16\right)\cdot 131^{4} + \left(23 a^{2} + 119 a + 65\right)\cdot 131^{5} + \left(76 a^{2} + 4 a + 55\right)\cdot 131^{6} + \left(79 a^{2} + 29 a + 29\right)\cdot 131^{7} + \left(81 a^{2} + 68 a + 4\right)\cdot 131^{8} + \left(105 a^{2} + 130 a + 72\right)\cdot 131^{9} +O(131^{10})\) |
Generators of the action on the roots $r_1, \ldots, r_{ 9 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 9 }$ | Character value |
$1$ | $1$ | $()$ | $12$ |
$9$ | $2$ | $(4,5)$ | $4$ |
$18$ | $2$ | $(2,4)(3,5)(7,9)$ | $2$ |
$27$ | $2$ | $(1,6)(2,3)(4,5)$ | $0$ |
$27$ | $2$ | $(2,3)(4,5)$ | $0$ |
$54$ | $2$ | $(1,2)(3,6)(4,5)(8,9)$ | $2$ |
$6$ | $3$ | $(1,6,8)$ | $0$ |
$8$ | $3$ | $(1,6,8)(2,3,9)(4,5,7)$ | $3$ |
$12$ | $3$ | $(1,6,8)(2,3,9)$ | $-3$ |
$72$ | $3$ | $(1,4,2)(3,6,5)(7,9,8)$ | $0$ |
$54$ | $4$ | $(2,4,3,5)(7,9)$ | $0$ |
$162$ | $4$ | $(1,4,6,5)(3,9)(7,8)$ | $0$ |
$36$ | $6$ | $(1,6,8)(2,4)(3,5)(7,9)$ | $2$ |
$36$ | $6$ | $(1,5,6,7,8,4)$ | $-1$ |
$36$ | $6$ | $(1,6,8)(4,5)$ | $-2$ |
$36$ | $6$ | $(1,6,8)(2,3,9)(4,5)$ | $1$ |
$54$ | $6$ | $(1,8,6)(2,3)(4,5)$ | $0$ |
$72$ | $6$ | $(1,6,8)(2,5,3,7,9,4)$ | $-1$ |
$108$ | $6$ | $(1,3,6,9,8,2)(4,5)$ | $-1$ |
$216$ | $6$ | $(1,4,3,6,5,2)(7,9,8)$ | $0$ |
$144$ | $9$ | $(1,5,3,6,7,9,8,4,2)$ | $0$ |
$108$ | $12$ | $(1,6,8)(2,4,3,5)(7,9)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.