Basic invariants
Dimension: | $12$ |
Group: | $S_3\wr S_3$ |
Conductor: | \(65577574924899328\)\(\medspace = 2^{10} \cdot 31^{4} \cdot 37^{5} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 9.3.14873341696.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 18T315 |
Parity: | even |
Projective image: | $S_3\wr S_3$ |
Projective field: | Galois closure of 9.3.14873341696.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 157 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 157 }$:
\( x^{3} + x + 152 \)
Roots:
$r_{ 1 }$ | $=$ |
\( a^{2} + 58 a + 46 + \left(155 a^{2} + 57 a + 75\right)\cdot 157 + \left(23 a + 81\right)\cdot 157^{2} + \left(6 a^{2} + 74 a + 10\right)\cdot 157^{3} + \left(52 a^{2} + 6 a + 125\right)\cdot 157^{4} + \left(95 a^{2} + 114 a + 16\right)\cdot 157^{5} + \left(110 a^{2} + 140 a + 23\right)\cdot 157^{6} + \left(86 a^{2} + 43 a + 100\right)\cdot 157^{7} + \left(97 a + 127\right)\cdot 157^{8} + \left(16 a^{2} + 123 a + 59\right)\cdot 157^{9} +O(157^{10})\)
$r_{ 2 }$ |
$=$ |
\( 24 a^{2} + 103 a + 85 + \left(93 a^{2} + 22 a + 133\right)\cdot 157 + \left(37 a^{2} + 6 a + 34\right)\cdot 157^{2} + \left(62 a^{2} + 86 a + 109\right)\cdot 157^{3} + \left(131 a^{2} + 104 a + 98\right)\cdot 157^{4} + \left(118 a^{2} + 70 a + 114\right)\cdot 157^{5} + \left(36 a^{2} + 50 a + 142\right)\cdot 157^{6} + \left(96 a^{2} + 106 a + 88\right)\cdot 157^{7} + \left(113 a^{2} + 121 a + 104\right)\cdot 157^{8} + \left(116 a^{2} + 136 a + 41\right)\cdot 157^{9} +O(157^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 37 a^{2} + 44 a + 70 + \left(18 a^{2} + 116 a + 36\right)\cdot 157 + \left(91 a^{2} + 27 a + 89\right)\cdot 157^{2} + \left(126 a^{2} + 57 a + 38\right)\cdot 157^{3} + \left(30 a^{2} + 82 a + 6\right)\cdot 157^{4} + \left(138 a^{2} + 88 a + 150\right)\cdot 157^{5} + \left(18 a^{2} + 101 a + 118\right)\cdot 157^{6} + \left(92 a^{2} + 124 a + 103\right)\cdot 157^{7} + \left(29 a^{2} + 96 a + 94\right)\cdot 157^{8} + \left(88 a^{2} + 44 a + 55\right)\cdot 157^{9} +O(157^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 55 a^{2} + 99 a + 28 + \left(59 a^{2} + 32 a + 153\right)\cdot 157 + \left(119 a^{2} + 156 a + 145\right)\cdot 157^{2} + \left(125 a^{2} + 62 a + 61\right)\cdot 157^{3} + \left(39 a^{2} + 58 a + 134\right)\cdot 157^{4} + \left(39 a^{2} + 136 a + 89\right)\cdot 157^{5} + \left(56 a^{2} + 80 a + 74\right)\cdot 157^{6} + \left(26 a^{2} + 83 a + 107\right)\cdot 157^{7} + \left(63 a^{2} + 31 a + 147\right)\cdot 157^{8} + \left(2 a^{2} + 111 a + 40\right)\cdot 157^{9} +O(157^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 105 a^{2} + 88 a + 9 + \left(62 a^{2} + 89 a + 103\right)\cdot 157 + \left(128 a^{2} + 74 a + 99\right)\cdot 157^{2} + \left(78 a^{2} + 147 a + 30\right)\cdot 157^{3} + \left(5 a^{2} + 64 a + 59\right)\cdot 157^{4} + \left(99 a^{2} + 45 a + 77\right)\cdot 157^{5} + \left(10 a^{2} + 92 a + 96\right)\cdot 157^{6} + \left(44 a^{2} + 12 a + 14\right)\cdot 157^{7} + \left(145 a^{2} + 111 a + 150\right)\cdot 157^{8} + \left(71 a^{2} + 77 a + 34\right)\cdot 157^{9} +O(157^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 119 a^{2} + 55 a + 20 + \left(140 a^{2} + 140 a + 118\right)\cdot 157 + \left(64 a^{2} + 105 a + 71\right)\cdot 157^{2} + \left(24 a^{2} + 25 a + 127\right)\cdot 157^{3} + \left(74 a^{2} + 68 a + 139\right)\cdot 157^{4} + \left(80 a^{2} + 111 a + 6\right)\cdot 157^{5} + \left(27 a^{2} + 71 a + 20\right)\cdot 157^{6} + \left(135 a^{2} + 145 a + 80\right)\cdot 157^{7} + \left(126 a^{2} + 119 a + 2\right)\cdot 157^{8} + \left(52 a^{2} + 145 a + 32\right)\cdot 157^{9} +O(157^{10})\)
| $r_{ 7 }$ |
$=$ |
\( 140 a^{2} + 42 a + 110 + \left(105 a^{2} + 124 a + 89\right)\cdot 157 + \left(20 a^{2} + 25 a + 23\right)\cdot 157^{2} + \left(77 a^{2} + 40 a + 119\right)\cdot 157^{3} + \left(27 a^{2} + 68 a + 81\right)\cdot 157^{4} + \left(29 a^{2} + 3 a + 2\right)\cdot 157^{5} + \left(57 a^{2} + 76 a + 104\right)\cdot 157^{6} + \left(138 a^{2} + 151 a + 64\right)\cdot 157^{7} + \left(128 a^{2} + 47 a + 62\right)\cdot 157^{8} + \left(87 a^{2} + 36 a + 22\right)\cdot 157^{9} +O(157^{10})\)
| $r_{ 8 }$ |
$=$ |
\( 150 a^{2} + 12 a + 12 + \left(114 a^{2} + 10 a + 148\right)\cdot 157 + \left(98 a^{2} + 125 a + 127\right)\cdot 157^{2} + \left(17 a^{2} + 30 a + 131\right)\cdot 157^{3} + \left(155 a^{2} + 141 a + 9\right)\cdot 157^{4} + \left(8 a^{2} + 82 a + 146\right)\cdot 157^{5} + \left(63 a^{2} + 30 a + 107\right)\cdot 157^{6} + \left(79 a^{2} + 56 a + 77\right)\cdot 157^{7} + \left(71 a^{2} + 144 a + 76\right)\cdot 157^{8} + \left(109 a^{2} + 140 a + 141\right)\cdot 157^{9} +O(157^{10})\)
| $r_{ 9 }$ |
$=$ |
\( 154 a^{2} + 127 a + 94 + \left(34 a^{2} + 34 a + 84\right)\cdot 157 + \left(66 a^{2} + 83 a + 110\right)\cdot 157^{2} + \left(109 a^{2} + 103 a + 155\right)\cdot 157^{3} + \left(111 a^{2} + 33 a + 129\right)\cdot 157^{4} + \left(18 a^{2} + 132 a + 23\right)\cdot 157^{5} + \left(90 a^{2} + 140 a + 97\right)\cdot 157^{6} + \left(86 a^{2} + 60 a + 147\right)\cdot 157^{7} + \left(105 a^{2} + 14 a + 18\right)\cdot 157^{8} + \left(82 a^{2} + 125 a + 42\right)\cdot 157^{9} +O(157^{10})\)
| |
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 values |
$c1$ | |||
$1$ | $1$ | $()$ | $12$ |
$9$ | $2$ | $(3,5)$ | $4$ |
$18$ | $2$ | $(1,3)(4,5)(7,8)$ | $2$ |
$27$ | $2$ | $(1,4)(2,6)(3,5)$ | $0$ |
$27$ | $2$ | $(2,6)(3,5)$ | $0$ |
$54$ | $2$ | $(1,4)(2,3)(5,6)(7,9)$ | $2$ |
$6$ | $3$ | $(2,6,9)$ | $0$ |
$8$ | $3$ | $(1,8,4)(2,9,6)(3,7,5)$ | $3$ |
$12$ | $3$ | $(2,9,6)(3,7,5)$ | $-3$ |
$72$ | $3$ | $(1,2,3)(4,6,5)(7,8,9)$ | $0$ |
$54$ | $4$ | $(2,5,6,3)(7,9)$ | $0$ |
$162$ | $4$ | $(1,4)(2,5,6,3)(7,9)$ | $0$ |
$36$ | $6$ | $(1,3)(2,6,9)(4,5)(7,8)$ | $2$ |
$36$ | $6$ | $(2,7,9,5,6,3)$ | $-1$ |
$36$ | $6$ | $(2,6,9)(3,5)$ | $-2$ |
$36$ | $6$ | $(1,4,8)(2,6,9)(3,5)$ | $1$ |
$54$ | $6$ | $(1,4)(2,9,6)(3,5)$ | $0$ |
$72$ | $6$ | $(1,7,8,5,4,3)(2,6,9)$ | $-1$ |
$108$ | $6$ | $(1,4)(2,7,9,5,6,3)$ | $-1$ |
$216$ | $6$ | $(1,2,5,4,6,3)(7,8,9)$ | $0$ |
$144$ | $9$ | $(1,2,7,8,9,5,4,6,3)$ | $0$ |
$108$ | $12$ | $(1,5,4,3)(2,6,9)(7,8)$ | $0$ |