Basic invariants
Dimension: | $6$ |
Group: | $C_3^2 : D_{6} $ |
Conductor: | \(984150000\)\(\medspace = 2^{4} \cdot 3^{9} \cdot 5^{5} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 9.1.295245000000.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 18T51 |
Parity: | odd |
Determinant: | 1.15.2t1.a.a |
Projective image: | $C_3^2:D_6$ |
Projective stem field: | Galois closure of 9.1.295245000000.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{9} + 9x^{7} - 3x^{6} + 27x^{5} - 18x^{4} + 30x^{3} - 27x^{2} + 9x - 11 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 139 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 139 }$: \( x^{3} + 6x + 137 \)
Roots:
$r_{ 1 }$ | $=$ | \( 4 + 57\cdot 139 + 8\cdot 139^{2} + 67\cdot 139^{3} + 64\cdot 139^{4} + 109\cdot 139^{5} + 8\cdot 139^{6} + 71\cdot 139^{7} + 55\cdot 139^{8} + 98\cdot 139^{9} +O(139^{10})\) |
$r_{ 2 }$ | $=$ | \( 45 + 120\cdot 139 + 3\cdot 139^{2} + 132\cdot 139^{3} + 9\cdot 139^{4} + 133\cdot 139^{5} + 101\cdot 139^{6} + 59\cdot 139^{7} + 132\cdot 139^{8} + 99\cdot 139^{9} +O(139^{10})\) |
$r_{ 3 }$ | $=$ | \( 90 + 100\cdot 139 + 126\cdot 139^{2} + 78\cdot 139^{3} + 64\cdot 139^{4} + 35\cdot 139^{5} + 28\cdot 139^{6} + 8\cdot 139^{7} + 90\cdot 139^{8} + 79\cdot 139^{9} +O(139^{10})\) |
$r_{ 4 }$ | $=$ | \( 19 a^{2} + 98 a + 76 + \left(103 a^{2} + 16 a + 134\right)\cdot 139 + \left(73 a^{2} + 80 a + 16\right)\cdot 139^{2} + \left(2 a^{2} + 111 a + 10\right)\cdot 139^{3} + \left(126 a^{2} + 32 a + 87\right)\cdot 139^{4} + \left(110 a^{2} + 88 a + 26\right)\cdot 139^{5} + \left(94 a^{2} + 43 a + 101\right)\cdot 139^{6} + \left(74 a^{2} + 120 a + 20\right)\cdot 139^{7} + \left(5 a^{2} + 30 a + 22\right)\cdot 139^{8} + \left(105 a^{2} + 56 a + 3\right)\cdot 139^{9} +O(139^{10})\) |
$r_{ 5 }$ | $=$ | \( 74 a^{2} + 13 a + 18 + \left(72 a^{2} + 108 a + 12\right)\cdot 139 + \left(92 a^{2} + 90 a + 92\right)\cdot 139^{2} + \left(124 a^{2} + 117 a + 81\right)\cdot 139^{3} + \left(38 a^{2} + 121 a + 16\right)\cdot 139^{4} + \left(20 a^{2} + 90 a + 81\right)\cdot 139^{5} + \left(51 a^{2} + 57 a + 65\right)\cdot 139^{6} + \left(96 a^{2} + 96 a + 107\right)\cdot 139^{7} + \left(101 a^{2} + 133 a + 128\right)\cdot 139^{8} + \left(131 a^{2} + 72 a + 109\right)\cdot 139^{9} +O(139^{10})\) |
$r_{ 6 }$ | $=$ | \( 87 a^{2} + 64 a + 70 + \left(53 a^{2} + 15 a + 75\right)\cdot 139 + \left(95 a^{2} + 126 a + 103\right)\cdot 139^{2} + \left(64 a^{2} + 61 a + 119\right)\cdot 139^{3} + \left(73 a^{2} + 35 a + 15\right)\cdot 139^{4} + \left(28 a^{2} + 123 a + 114\right)\cdot 139^{5} + \left(50 a^{2} + 9 a + 61\right)\cdot 139^{6} + \left(136 a^{2} + 2 a + 128\right)\cdot 139^{7} + \left(43 a^{2} + 100 a + 36\right)\cdot 139^{8} + \left(79 a^{2} + 21 a + 39\right)\cdot 139^{9} +O(139^{10})\) |
$r_{ 7 }$ | $=$ | \( 117 a^{2} + 62 a + 51 + \left(12 a^{2} + 15 a + 51\right)\cdot 139 + \left(90 a^{2} + 61 a + 82\right)\cdot 139^{2} + \left(88 a^{2} + 98 a + 76\right)\cdot 139^{3} + \left(26 a^{2} + 120 a + 106\right)\cdot 139^{4} + \left(90 a^{2} + 63 a + 82\right)\cdot 139^{5} + \left(37 a^{2} + 71 a + 11\right)\cdot 139^{6} + \left(45 a^{2} + 40 a + 42\right)\cdot 139^{7} + \left(132 a^{2} + 44 a + 112\right)\cdot 139^{8} + \left(66 a^{2} + 44 a + 128\right)\cdot 139^{9} +O(139^{10})\) |
$r_{ 8 }$ | $=$ | \( 124 a^{2} + 127 a + 79 + \left(135 a^{2} + 85 a + 126\right)\cdot 139 + \left(123 a^{2} + 48 a + 78\right)\cdot 139^{2} + \left(53 a^{2} + 51 a + 76\right)\cdot 139^{3} + \left(103 a^{2} + 135 a + 135\right)\cdot 139^{4} + \left(23 a^{2} + 14 a + 94\right)\cdot 139^{5} + \left(27 a^{2} + 23 a + 108\right)\cdot 139^{6} + \left(66 a^{2} + 40 a + 125\right)\cdot 139^{7} + \left(99 a^{2} + 81 a + 119\right)\cdot 139^{8} + \left(77 a^{2} + 119 a + 32\right)\cdot 139^{9} +O(139^{10})\) |
$r_{ 9 }$ | $=$ | \( 135 a^{2} + 53 a + 123 + \left(38 a^{2} + 36 a + 16\right)\cdot 139 + \left(80 a^{2} + 10 a + 43\right)\cdot 139^{2} + \left(82 a^{2} + 115 a + 52\right)\cdot 139^{3} + \left(48 a^{2} + 109 a + 55\right)\cdot 139^{4} + \left(4 a^{2} + 35 a + 17\right)\cdot 139^{5} + \left(17 a^{2} + 72 a + 68\right)\cdot 139^{6} + \left(137 a^{2} + 117 a + 131\right)\cdot 139^{7} + \left(33 a^{2} + 26 a + 135\right)\cdot 139^{8} + \left(95 a^{2} + 102 a + 102\right)\cdot 139^{9} +O(139^{10})\) |
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 |
$1$ | $1$ | $()$ | $6$ |
$9$ | $2$ | $(2,3)(4,9)(6,7)$ | $0$ |
$9$ | $2$ | $(4,5)(6,9)(7,8)$ | $-2$ |
$9$ | $2$ | $(1,9)(2,8)(3,4)(5,7)$ | $0$ |
$2$ | $3$ | $(1,2,3)(4,8,9)(5,7,6)$ | $-3$ |
$6$ | $3$ | $(1,3,2)(5,7,6)$ | $0$ |
$6$ | $3$ | $(1,5,8)(2,7,9)(3,6,4)$ | $0$ |
$12$ | $3$ | $(1,4,5)(2,8,7)(3,9,6)$ | $0$ |
$18$ | $6$ | $(1,8,5)(2,4,7,3,9,6)$ | $0$ |
$18$ | $6$ | $(1,3,2)(4,7,8,6,9,5)$ | $1$ |
$18$ | $6$ | $(1,8,3,9,2,4)(5,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.