Basic invariants
| Dimension: | $12$ |
| Group: | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ |
| Conductor: | \(8569121343897600000\)\(\medspace = 2^{18} \cdot 3^{21} \cdot 5^{5} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.5.1836660096000.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T219 |
| Parity: | even |
| Determinant: | 1.60.2t1.a.a |
| Projective image: | $C_3^3:S_4$ |
| Projective stem field: | Galois closure of 9.5.1836660096000.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 3x^{8} - 9x^{7} + 17x^{6} + 60x^{5} - 36x^{4} - 160x^{3} - 24x^{2} + 96x + 32 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 191 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 191 }$:
\( x^{3} + 4x + 172 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 40 + 154\cdot 191 + 114\cdot 191^{2} + 191^{3} + 2\cdot 191^{4} + 5\cdot 191^{5} + 190\cdot 191^{6} + 154\cdot 191^{7} + 14\cdot 191^{8} + 77\cdot 191^{9} +O(191^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 114 + 15\cdot 191 + 131\cdot 191^{2} + 180\cdot 191^{3} + 183\cdot 191^{4} + 122\cdot 191^{5} + 68\cdot 191^{6} + 114\cdot 191^{7} + 172\cdot 191^{8} + 124\cdot 191^{9} +O(191^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 139 + 19\cdot 191 + 174\cdot 191^{2} + 119\cdot 191^{3} + 92\cdot 191^{4} + 182\cdot 191^{5} + 101\cdot 191^{6} + 46\cdot 191^{7} + 8\cdot 191^{8} + 90\cdot 191^{9} +O(191^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 74 a^{2} + 124 a + 88 + \left(72 a^{2} + 92 a + 66\right)\cdot 191 + \left(43 a^{2} + 164 a + 176\right)\cdot 191^{2} + \left(128 a^{2} + 189 a + 23\right)\cdot 191^{3} + \left(30 a^{2} + 151 a + 38\right)\cdot 191^{4} + \left(104 a^{2} + 188 a + 128\right)\cdot 191^{5} + \left(6 a^{2} + 120 a + 155\right)\cdot 191^{6} + \left(109 a^{2} + 48 a + 44\right)\cdot 191^{7} + \left(123 a^{2} + 133 a + 146\right)\cdot 191^{8} + \left(49 a^{2} + 19 a + 77\right)\cdot 191^{9} +O(191^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 76 a^{2} + 105 a + 88 + \left(67 a^{2} + 152 a + 179\right)\cdot 191 + \left(6 a^{2} + 92 a + 70\right)\cdot 191^{2} + \left(122 a^{2} + 120 a + 33\right)\cdot 191^{3} + \left(82 a^{2} + 40 a + 171\right)\cdot 191^{4} + \left(166 a^{2} + 158 a + 107\right)\cdot 191^{5} + \left(128 a^{2} + 172 a + 21\right)\cdot 191^{6} + \left(181 a^{2} + 35 a + 52\right)\cdot 191^{7} + \left(159 a^{2} + 29 a + 35\right)\cdot 191^{8} + \left(21 a^{2} + 98 a + 79\right)\cdot 191^{9} +O(191^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 121 a^{2} + 85 a + 17 + \left(124 a^{2} + 117 a + 141\right)\cdot 191 + \left(47 a^{2} + 133 a + 53\right)\cdot 191^{2} + \left(71 a^{2} + 18 a + 25\right)\cdot 191^{3} + \left(29 a^{2} + 31 a + 29\right)\cdot 191^{4} + \left(178 a^{2} + 60 a + 139\right)\cdot 191^{5} + \left(10 a^{2} + 182 a + 152\right)\cdot 191^{6} + \left(3 a^{2} + 176 a + 148\right)\cdot 191^{7} + \left(71 a^{2} + 41 a + 52\right)\cdot 191^{8} + \left(104 a^{2} + 151 a + 108\right)\cdot 191^{9} +O(191^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 124 a^{2} + 55 a + 94 + \left(34 a^{2} + 5 a + 29\right)\cdot 191 + \left(94 a^{2} + 73 a + 57\right)\cdot 191^{2} + \left(160 a^{2} + 105 a + 46\right)\cdot 191^{3} + \left(189 a^{2} + 12 a + 144\right)\cdot 191^{4} + \left(146 a^{2} + 34 a + 178\right)\cdot 191^{5} + \left(35 a^{2} + 67 a + 169\right)\cdot 191^{6} + \left(8 a^{2} + 168 a + 157\right)\cdot 191^{7} + \left(104 a^{2} + 99 a + 157\right)\cdot 191^{8} + \left(27 a^{2} + 21 a + 18\right)\cdot 191^{9} +O(191^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 184 a^{2} + 12 a + 63 + \left(83 a^{2} + 93 a + 97\right)\cdot 191 + \left(53 a^{2} + 144 a + 139\right)\cdot 191^{2} + \left(93 a^{2} + 86 a + 121\right)\cdot 191^{3} + \left(161 a^{2} + 26 a + 68\right)\cdot 191^{4} + \left(130 a^{2} + 159 a + 8\right)\cdot 191^{5} + \left(148 a^{2} + 2 a + 89\right)\cdot 191^{6} + \left(73 a^{2} + 165 a + 14\right)\cdot 191^{7} + \left(154 a^{2} + 148 a + 101\right)\cdot 191^{8} + \left(113 a^{2} + 149 a + 121\right)\cdot 191^{9} +O(191^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 185 a^{2} + a + 124 + \left(189 a^{2} + 112 a + 60\right)\cdot 191 + \left(136 a^{2} + 155 a + 37\right)\cdot 191^{2} + \left(188 a^{2} + 51 a + 20\right)\cdot 191^{3} + \left(78 a^{2} + 119 a + 34\right)\cdot 191^{4} + \left(37 a^{2} + 163 a + 82\right)\cdot 191^{5} + \left(51 a^{2} + 26 a + 5\right)\cdot 191^{6} + \left(6 a^{2} + 169 a + 30\right)\cdot 191^{7} + \left(151 a^{2} + 119 a + 75\right)\cdot 191^{8} + \left(64 a^{2} + 132 a + 66\right)\cdot 191^{9} +O(191^{10})\)
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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 |
| $1$ | $1$ | $()$ | $12$ |
| $18$ | $2$ | $(1,8)(2,4)(3,7)$ | $2$ |
| $27$ | $2$ | $(2,3)(4,8)$ | $0$ |
| $4$ | $3$ | $(1,3,2)(4,8,7)(5,6,9)$ | $3$ |
| $4$ | $3$ | $(1,2,3)(4,7,8)(5,9,6)$ | $3$ |
| $6$ | $3$ | $(5,6,9)$ | $0$ |
| $12$ | $3$ | $(1,3,2)(4,7,8)$ | $-3$ |
| $72$ | $3$ | $(1,8,6)(2,7,9)(3,4,5)$ | $0$ |
| $162$ | $4$ | $(1,4,2,8)(3,7)(5,6)$ | $0$ |
| $18$ | $6$ | $(1,8)(2,4)(3,7)(5,6,9)$ | $2$ |
| $18$ | $6$ | $(1,8)(2,4)(3,7)(5,9,6)$ | $2$ |
| $36$ | $6$ | $(1,2,3)(4,9,7,5,8,6)$ | $-1$ |
| $36$ | $6$ | $(1,3,2)(4,6,8,5,7,9)$ | $-1$ |
| $36$ | $6$ | $(4,9,7,5,8,6)$ | $-1$ |
| $54$ | $6$ | $(2,3)(4,8)(5,9,6)$ | $0$ |
| $72$ | $9$ | $(1,4,9,3,8,5,2,7,6)$ | $0$ |
| $72$ | $9$ | $(1,9,8,2,6,4,3,5,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.