Basic invariants
| Dimension: | $8$ |
| Group: | $S_3\wr S_3$ |
| Conductor: | \(9069133824\)\(\medspace = 2^{20} \cdot 3^{2} \cdot 31^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 9.1.26357170176.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T213 |
| Parity: | even |
| Projective image: | $S_3\wr S_3$ |
| Projective field: | Galois closure of 9.1.26357170176.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 181 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 181 }$:
\( x^{3} + 6x + 179 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 82 + 160\cdot 181 + 105\cdot 181^{2} + 111\cdot 181^{3} + 118\cdot 181^{4} + 124\cdot 181^{5} + 100\cdot 181^{6} + 9\cdot 181^{7} + 63\cdot 181^{8} + 164\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 121 + 146\cdot 181 + 73\cdot 181^{2} + 25\cdot 181^{3} + 99\cdot 181^{4} + 39\cdot 181^{5} + 118\cdot 181^{6} + 60\cdot 181^{7} + 113\cdot 181^{8} + 160\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 159 + 54\cdot 181 + 181^{2} + 44\cdot 181^{3} + 144\cdot 181^{4} + 16\cdot 181^{5} + 143\cdot 181^{6} + 110\cdot 181^{7} + 4\cdot 181^{8} + 37\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 22 a^{2} + 91 a + 88 + \left(143 a^{2} + 126 a + 29\right)\cdot 181 + \left(55 a^{2} + 20 a + 42\right)\cdot 181^{2} + \left(2 a^{2} + 142 a + 9\right)\cdot 181^{3} + \left(137 a^{2} + 81 a + 5\right)\cdot 181^{4} + \left(37 a^{2} + 134 a + 151\right)\cdot 181^{5} + \left(138 a^{2} + 140 a + 9\right)\cdot 181^{6} + \left(103 a^{2} + 44 a + 53\right)\cdot 181^{7} + \left(32 a^{2} + 163 a + 130\right)\cdot 181^{8} + \left(177 a^{2} + 150 a + 165\right)\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 57 a^{2} + 17 a + 47 + \left(25 a^{2} + 43 a + 101\right)\cdot 181 + \left(175 a^{2} + 128 a + 157\right)\cdot 181^{2} + \left(43 a^{2} + 56 a + 175\right)\cdot 181^{3} + \left(31 a^{2} + 93 a + 124\right)\cdot 181^{4} + \left(62 a^{2} + 113 a + 67\right)\cdot 181^{5} + \left(59 a^{2} + 31 a + 56\right)\cdot 181^{6} + \left(159 a^{2} + 157 a + 94\right)\cdot 181^{7} + \left(137 a^{2} + 59 a + 8\right)\cdot 181^{8} + \left(172 a^{2} + 115 a + 148\right)\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 92 a^{2} + 99 a + 6 + \left(143 a^{2} + 113 a + 31\right)\cdot 181 + \left(116 a^{2} + 104 a + 105\right)\cdot 181^{2} + \left(36 a^{2} + 2 a + 146\right)\cdot 181^{3} + \left(60 a^{2} + 135 a + 59\right)\cdot 181^{4} + \left(101 a^{2} + 86 a + 43\right)\cdot 181^{5} + \left(32 a^{2} + 156 a + 130\right)\cdot 181^{6} + \left(126 a^{2} + 88 a + 142\right)\cdot 181^{7} + \left(163 a^{2} + 175 a + 111\right)\cdot 181^{8} + \left(18 a^{2} + 76 a + 75\right)\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 99 a^{2} + 19 a + 34 + \left(140 a^{2} + 137 a + 19\right)\cdot 181 + \left(6 a^{2} + 43 a + 27\right)\cdot 181^{2} + \left(91 a^{2} + 104 a + 2\right)\cdot 181^{3} + \left(119 a^{2} + 111 a + 116\right)\cdot 181^{4} + \left(95 a^{2} + 127 a + 20\right)\cdot 181^{5} + \left(141 a^{2} + 29 a + 23\right)\cdot 181^{6} + \left(84 a^{2} + a + 158\right)\cdot 181^{7} + \left(77 a^{2} + 86 a + 128\right)\cdot 181^{8} + \left(117 a^{2} + 58 a + 107\right)\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 102 a^{2} + 73 a + 46 + \left(12 a^{2} + 11 a + 50\right)\cdot 181 + \left(131 a^{2} + 32 a + 162\right)\cdot 181^{2} + \left(134 a^{2} + 163 a + 176\right)\cdot 181^{3} + \left(12 a^{2} + 5 a + 50\right)\cdot 181^{4} + \left(81 a^{2} + 114 a + 143\right)\cdot 181^{5} + \left(164 a^{2} + 8 a + 114\right)\cdot 181^{6} + \left(98 a^{2} + 160 a + 33\right)\cdot 181^{7} + \left(10 a^{2} + 138 a + 42\right)\cdot 181^{8} + \left(12 a^{2} + 95 a + 48\right)\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 171 a^{2} + 63 a + 141 + \left(77 a^{2} + 111 a + 130\right)\cdot 181 + \left(57 a^{2} + 32 a + 48\right)\cdot 181^{2} + \left(53 a^{2} + 74 a + 32\right)\cdot 181^{3} + \left(a^{2} + 115 a + 5\right)\cdot 181^{4} + \left(165 a^{2} + 147 a + 117\right)\cdot 181^{5} + \left(6 a^{2} + 175 a + 27\right)\cdot 181^{6} + \left(151 a^{2} + 90 a + 61\right)\cdot 181^{7} + \left(120 a^{2} + 100 a + 121\right)\cdot 181^{8} + \left(44 a^{2} + 45 a + 178\right)\cdot 181^{9} +O(181^{10})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $8$ |
| $9$ | $2$ | $(1,2)$ | $0$ |
| $18$ | $2$ | $(1,6)(2,7)(3,9)$ | $4$ |
| $27$ | $2$ | $(1,2)(4,5)(6,7)$ | $0$ |
| $27$ | $2$ | $(1,2)(4,5)$ | $0$ |
| $54$ | $2$ | $(1,4)(2,5)(3,8)(6,7)$ | $0$ |
| $6$ | $3$ | $(4,5,8)$ | $-4$ |
| $8$ | $3$ | $(1,3,2)(4,8,5)(6,9,7)$ | $-1$ |
| $12$ | $3$ | $(1,3,2)(4,8,5)$ | $2$ |
| $72$ | $3$ | $(1,6,4)(2,7,5)(3,9,8)$ | $2$ |
| $54$ | $4$ | $(1,4,2,5)(3,8)$ | $0$ |
| $162$ | $4$ | $(1,4,2,5)(3,8)(6,7)$ | $0$ |
| $36$ | $6$ | $(1,6)(2,7)(3,9)(4,5,8)$ | $-2$ |
| $36$ | $6$ | $(1,4,3,8,2,5)$ | $-2$ |
| $36$ | $6$ | $(1,2)(4,5,8)$ | $0$ |
| $36$ | $6$ | $(1,2)(4,5,8)(6,7,9)$ | $0$ |
| $54$ | $6$ | $(1,2)(4,8,5)(6,7)$ | $0$ |
| $72$ | $6$ | $(1,6,3,9,2,7)(4,5,8)$ | $1$ |
| $108$ | $6$ | $(1,4,3,8,2,5)(6,7)$ | $0$ |
| $216$ | $6$ | $(1,6,4,2,7,5)(3,9,8)$ | $0$ |
| $144$ | $9$ | $(1,6,4,3,9,8,2,7,5)$ | $-1$ |
| $108$ | $12$ | $(1,6,2,7)(3,9)(4,5,8)$ | $0$ |