Basic invariants
| Dimension: | $12$ |
| Group: | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ |
| Conductor: | \(372173693419261107\)\(\medspace = 3^{15} \cdot 11^{10} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.1.113919098077521.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T218 |
| Parity: | odd |
| Determinant: | 1.3.2t1.a.a |
| Projective image: | $C_3^3:S_4$ |
| Projective stem field: | Galois closure of 9.1.113919098077521.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 3x^{8} + 15x^{7} - 30x^{6} + 15x^{5} - 27x^{4} - 93x^{3} + 51x^{2} + 48x - 25 \)
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The roots of $f$ are computed in an extension of $\Q_{ 193 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 193 }$:
\( x^{3} + x + 188 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 55 + 147\cdot 193 + 122\cdot 193^{2} + 40\cdot 193^{3} + 172\cdot 193^{4} + 130\cdot 193^{5} + 42\cdot 193^{6} + 129\cdot 193^{7} + 70\cdot 193^{8} + 7\cdot 193^{9} +O(193^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 113 + 82\cdot 193 + 65\cdot 193^{2} + 144\cdot 193^{3} + 5\cdot 193^{4} + 160\cdot 193^{5} + 87\cdot 193^{6} + 150\cdot 193^{7} + 153\cdot 193^{8} + 100\cdot 193^{9} +O(193^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 131 + 3\cdot 193 + 45\cdot 193^{2} + 143\cdot 193^{3} + 114\cdot 193^{4} + 169\cdot 193^{5} + 176\cdot 193^{6} + 123\cdot 193^{7} + 56\cdot 193^{8} + 125\cdot 193^{9} +O(193^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 15 a^{2} + 135 a + 59 + \left(174 a^{2} + 19 a + 63\right)\cdot 193 + \left(49 a^{2} + 187 a + 104\right)\cdot 193^{2} + \left(25 a + 131\right)\cdot 193^{3} + \left(146 a^{2} + 165 a + 185\right)\cdot 193^{4} + \left(123 a^{2} + 20 a + 124\right)\cdot 193^{5} + \left(36 a^{2} + 189 a + 97\right)\cdot 193^{6} + \left(45 a^{2} + 159 a + 76\right)\cdot 193^{7} + \left(150 a^{2} + 29 a + 76\right)\cdot 193^{8} + \left(18 a^{2} + 188 a + 103\right)\cdot 193^{9} +O(193^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 73 a^{2} + 56 a + 162 + \left(3 a^{2} + 113 a + 13\right)\cdot 193 + \left(3 a^{2} + 178 a + 73\right)\cdot 193^{2} + \left(58 a^{2} + 9 a + 41\right)\cdot 193^{3} + \left(86 a^{2} + 12 a + 17\right)\cdot 193^{4} + \left(61 a^{2} + 80 a + 19\right)\cdot 193^{5} + \left(161 a^{2} + 48 a + 52\right)\cdot 193^{6} + \left(168 a^{2} + 168 a + 30\right)\cdot 193^{7} + \left(179 a^{2} + 5 a + 96\right)\cdot 193^{8} + \left(8 a^{2} + 12 a + 32\right)\cdot 193^{9} +O(193^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 105 a^{2} + 2 a + 119 + \left(15 a^{2} + 60 a + 150\right)\cdot 193 + \left(140 a^{2} + 20 a + 35\right)\cdot 193^{2} + \left(134 a^{2} + 157 a + 28\right)\cdot 193^{3} + \left(153 a^{2} + 15 a + 62\right)\cdot 193^{4} + \left(7 a^{2} + 92 a + 176\right)\cdot 193^{5} + \left(188 a^{2} + 148 a + 69\right)\cdot 193^{6} + \left(171 a^{2} + 57 a + 32\right)\cdot 193^{7} + \left(55 a^{2} + 157 a + 142\right)\cdot 193^{8} + \left(165 a^{2} + 185 a + 136\right)\cdot 193^{9} +O(193^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 120 a^{2} + 128 a + 61 + \left(78 a^{2} + 115 a + 27\right)\cdot 193 + \left(32 a^{2} + 105 a + 130\right)\cdot 193^{2} + \left(27 a^{2} + 104 a + 34\right)\cdot 193^{3} + \left(139 a^{2} + 141 a + 164\right)\cdot 193^{4} + \left(64 a^{2} + 104 a + 168\right)\cdot 193^{5} + \left(162 a^{2} + 44 a + 189\right)\cdot 193^{6} + \left(29 a^{2} + 189 a + 31\right)\cdot 193^{7} + \left(185 a^{2} + 113 a + 182\right)\cdot 193^{8} + \left(95 a^{2} + 13 a + 23\right)\cdot 193^{9} +O(193^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 133 a^{2} + 111 a + 134 + \left(34 a^{2} + 76 a + 126\right)\cdot 193 + \left(11 a^{2} + 109 a + 51\right)\cdot 193^{2} + \left(25 a^{2} + 88 a + 33\right)\cdot 193^{3} + \left(134 a^{2} + 90 a + 32\right)\cdot 193^{4} + \left(170 a^{2} + 74 a + 175\right)\cdot 193^{5} + \left(130 a^{2} + 103 a + 168\right)\cdot 193^{6} + \left(42 a^{2} + 24 a + 104\right)\cdot 193^{7} + \left(87 a^{2} + 148 a + 52\right)\cdot 193^{8} + \left(26 a^{2} + 140 a + 106\right)\cdot 193^{9} +O(193^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 133 a^{2} + 147 a + 134 + \left(79 a^{2} + 156\right)\cdot 193 + \left(149 a^{2} + 171 a + 143\right)\cdot 193^{2} + \left(140 a^{2} + 192 a + 174\right)\cdot 193^{3} + \left(112 a^{2} + 153 a + 17\right)\cdot 193^{4} + \left(150 a^{2} + 13 a + 33\right)\cdot 193^{5} + \left(92 a^{2} + 45 a + 79\right)\cdot 193^{6} + \left(120 a^{2} + 172 a + 92\right)\cdot 193^{7} + \left(113 a^{2} + 123 a + 134\right)\cdot 193^{8} + \left(70 a^{2} + 38 a + 135\right)\cdot 193^{9} +O(193^{10})\)
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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 |
| $1$ | $1$ | $()$ | $12$ |
| $27$ | $2$ | $(4,5)(7,8)$ | $0$ |
| $54$ | $2$ | $(1,7)(2,8)(3,9)(4,5)$ | $2$ |
| $6$ | $3$ | $(4,6,5)$ | $0$ |
| $8$ | $3$ | $(1,2,3)(4,5,6)(7,8,9)$ | $3$ |
| $12$ | $3$ | $(4,6,5)(7,9,8)$ | $-3$ |
| $72$ | $3$ | $(1,7,4)(2,8,5)(3,9,6)$ | $0$ |
| $54$ | $4$ | $(4,8,5,7)(6,9)$ | $0$ |
| $54$ | $6$ | $(1,2)(4,5,6)(7,8)$ | $0$ |
| $108$ | $6$ | $(1,2)(4,9,6,8,5,7)$ | $-1$ |
| $72$ | $9$ | $(1,7,4,2,8,5,3,9,6)$ | $0$ |
| $72$ | $9$ | $(1,7,4,3,9,6,2,8,5)$ | $0$ |
| $54$ | $12$ | $(1,8,2,7)(3,9)(4,6,5)$ | $0$ |
| $54$ | $12$ | $(1,8,2,7)(3,9)(4,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.