Basic invariants
Dimension: | $8$ |
Group: | $S_3\wr S_3$ |
Conductor: | \(1507335685696\)\(\medspace = 2^{6} \cdot 43^{4} \cdot 83^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 9.1.1573557824.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 12T213 |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $S_3\wr S_3$ |
Projective stem field: | Galois closure of 9.1.1573557824.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{9} - 4x^{8} + 8x^{7} - 9x^{6} + 8x^{5} - 6x^{4} + 5x^{3} - 2x^{2} + 2x - 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 109 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 109 }$: \( x^{3} + x + 103 \)
Roots:
$r_{ 1 }$ | $=$ | \( 30 a^{2} + 50 a + 43 + \left(47 a^{2} + 29 a + 81\right)\cdot 109 + \left(22 a^{2} + 13\right)\cdot 109^{2} + \left(13 a^{2} + 52 a + 44\right)\cdot 109^{3} + \left(53 a^{2} + 37 a + 37\right)\cdot 109^{4} + \left(96 a^{2} + 101 a + 19\right)\cdot 109^{5} + \left(24 a^{2} + 44 a + 92\right)\cdot 109^{6} + \left(70 a^{2} + 104 a + 61\right)\cdot 109^{7} + \left(2 a^{2} + 64 a + 61\right)\cdot 109^{8} + \left(96 a^{2} + 99 a + 45\right)\cdot 109^{9} +O(109^{10})\) |
$r_{ 2 }$ | $=$ | \( 30 a^{2} + 63 a + 78 + \left(92 a^{2} + 9 a + 104\right)\cdot 109 + \left(89 a^{2} + 83 a + 18\right)\cdot 109^{2} + \left(37 a^{2} + 4 a + 37\right)\cdot 109^{3} + \left(73 a^{2} + 71 a + 22\right)\cdot 109^{4} + \left(45 a^{2} + 88 a + 22\right)\cdot 109^{5} + \left(32 a^{2} + 30 a + 89\right)\cdot 109^{6} + \left(22 a^{2} + 108 a + 64\right)\cdot 109^{7} + \left(40 a^{2} + 62 a + 77\right)\cdot 109^{8} + \left(41 a^{2} + 6 a + 72\right)\cdot 109^{9} +O(109^{10})\) |
$r_{ 3 }$ | $=$ | \( 32 a^{2} + 65 a + 8 + \left(22 a^{2} + 24 a + 101\right)\cdot 109 + \left(97 a^{2} + 53 a + 99\right)\cdot 109^{2} + \left(96 a^{2} + 62 a + 99\right)\cdot 109^{3} + \left(44 a^{2} + 21 a + 31\right)\cdot 109^{4} + \left(33 a^{2} + 28 a + 86\right)\cdot 109^{5} + \left(94 a^{2} + 54 a + 65\right)\cdot 109^{6} + \left(76 a^{2} + 90 a + 102\right)\cdot 109^{7} + \left(70 a^{2} + 46 a + 106\right)\cdot 109^{8} + \left(75 a^{2} + 5 a + 31\right)\cdot 109^{9} +O(109^{10})\) |
$r_{ 4 }$ | $=$ | \( 46 a^{2} + 101 a + 60 + \left(98 a^{2} + 47 a + 81\right)\cdot 109 + \left(70 a^{2} + 37 a + 16\right)\cdot 109^{2} + \left(100 a^{2} + 53 a + 56\right)\cdot 109^{3} + \left(8 a^{2} + 99 a + 30\right)\cdot 109^{4} + \left(84 a^{2} + 31 a\right)\cdot 109^{5} + \left(47 a^{2} + 73 a + 34\right)\cdot 109^{6} + \left(47 a^{2} + 47 a + 39\right)\cdot 109^{7} + \left(16 a^{2} + 99 a + 9\right)\cdot 109^{8} + \left(60 a^{2} + 88 a + 86\right)\cdot 109^{9} +O(109^{10})\) |
$r_{ 5 }$ | $=$ | \( 47 a^{2} + 103 a + 18 + \left(39 a^{2} + 54 a + 76\right)\cdot 109 + \left(98 a^{2} + 55 a + 100\right)\cdot 109^{2} + \left(107 a^{2} + 103 a + 70\right)\cdot 109^{3} + \left(10 a^{2} + 49 a + 45\right)\cdot 109^{4} + \left(88 a^{2} + 88 a + 86\right)\cdot 109^{5} + \left(98 a^{2} + 9 a + 68\right)\cdot 109^{6} + \left(70 a^{2} + 23 a + 98\right)\cdot 109^{7} + \left(35 a^{2} + 106 a + 10\right)\cdot 109^{8} + \left(46 a^{2} + 3 a + 85\right)\cdot 109^{9} +O(109^{10})\) |
$r_{ 6 }$ | $=$ | \( 80 a^{2} + 12 a + 75 + \left(41 a^{2} + 84 a + 34\right)\cdot 109 + \left(51 a^{2} + 17 a + 102\right)\cdot 109^{2} + \left(93 a^{2} + 20 a + 37\right)\cdot 109^{3} + \left(18 a^{2} + 69 a + 22\right)\cdot 109^{4} + \left(66 a^{2} + 94 a + 72\right)\cdot 109^{5} + \left(2 a^{2} + 28 a + 105\right)\cdot 109^{6} + \left(48 a^{2} + 8 a + 81\right)\cdot 109^{7} + \left(11 a^{2} + 72 a + 94\right)\cdot 109^{8} + \left(108 a^{2} + 79 a + 80\right)\cdot 109^{9} +O(109^{10})\) |
$r_{ 7 }$ | $=$ | \( 84 a^{2} + 64 a + 49 + \left(66 a^{2} + 19 a + 60\right)\cdot 109 + \left(7 a^{2} + 98 a + 83\right)\cdot 109^{2} + \left(18 a^{2} + 49 a + 73\right)\cdot 109^{3} + \left(105 a^{2} + 28 a + 94\right)\cdot 109^{4} + \left(89 a^{2} + 23 a + 76\right)\cdot 109^{5} + \left(26 a^{2} + 53 a + 92\right)\cdot 109^{6} + \left(26 a^{2} + 14 a + 97\right)\cdot 109^{7} + \left(70 a^{2} + 68 a + 8\right)\cdot 109^{8} + \left(57 a^{2} + 57 a + 48\right)\cdot 109^{9} +O(109^{10})\) |
$r_{ 8 }$ | $=$ | \( 88 a^{2} + 53 a + 88 + \left(52 a^{2} + 41 a + 14\right)\cdot 109 + \left(30 a^{2} + 82 a + 26\right)\cdot 109^{2} + \left(99 a^{2} + 5 a + 55\right)\cdot 109^{3} + \left(103 a^{2} + 90 a + 57\right)\cdot 109^{4} + \left(43 a^{2} + 53 a + 82\right)\cdot 109^{5} + \left(34 a^{2} + 91 a + 97\right)\cdot 109^{6} + \left(35 a^{2} + 46 a + 103\right)\cdot 109^{7} + \left(22 a^{2} + 50 a + 85\right)\cdot 109^{8} + \left(100 a^{2} + 71 a + 3\right)\cdot 109^{9} +O(109^{10})\) |
$r_{ 9 }$ | $=$ | \( 108 a^{2} + 34 a + 21 + \left(83 a^{2} + 15 a + 99\right)\cdot 109 + \left(76 a^{2} + 8 a + 82\right)\cdot 109^{2} + \left(86 a^{2} + 84 a + 69\right)\cdot 109^{3} + \left(16 a^{2} + 77 a + 93\right)\cdot 109^{4} + \left(106 a^{2} + 34 a + 98\right)\cdot 109^{5} + \left(73 a^{2} + 49 a + 7\right)\cdot 109^{6} + \left(38 a^{2} + 101 a + 3\right)\cdot 109^{7} + \left(57 a^{2} + 82 a + 89\right)\cdot 109^{8} + \left(68 a^{2} + 22 a + 90\right)\cdot 109^{9} +O(109^{10})\) |
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 |
$1$ | $1$ | $()$ | $8$ |
$9$ | $2$ | $(3,4)$ | $0$ |
$18$ | $2$ | $(1,3)(4,8)(6,9)$ | $4$ |
$27$ | $2$ | $(1,8)(2,5)(3,4)$ | $0$ |
$27$ | $2$ | $(2,5)(3,4)$ | $0$ |
$54$ | $2$ | $(1,2)(3,4)(5,8)(7,9)$ | $0$ |
$6$ | $3$ | $(2,5,7)$ | $-4$ |
$8$ | $3$ | $(1,9,8)(2,7,5)(3,6,4)$ | $-1$ |
$12$ | $3$ | $(1,8,9)(2,5,7)$ | $2$ |
$72$ | $3$ | $(1,2,3)(4,8,5)(6,9,7)$ | $2$ |
$54$ | $4$ | $(2,4,5,3)(6,7)$ | $0$ |
$162$ | $4$ | $(2,4,5,3)(6,7)(8,9)$ | $0$ |
$36$ | $6$ | $(1,3)(2,5,7)(4,8)(6,9)$ | $-2$ |
$36$ | $6$ | $(2,6,7,4,5,3)$ | $-2$ |
$36$ | $6$ | $(2,5,7)(3,4)$ | $0$ |
$36$ | $6$ | $(1,8,9)(2,5,7)(3,4)$ | $0$ |
$54$ | $6$ | $(1,9,8)(2,5)(3,4)$ | $0$ |
$72$ | $6$ | $(1,4,8,6,9,3)(2,5,7)$ | $1$ |
$108$ | $6$ | $(1,2,8,5,9,7)(3,4)$ | $0$ |
$216$ | $6$ | $(1,2,4,8,5,3)(6,9,7)$ | $0$ |
$144$ | $9$ | $(1,2,6,9,7,4,8,5,3)$ | $-1$ |
$108$ | $12$ | $(1,8,9)(2,4,5,3)(6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.