Basic invariants
| Dimension: | $4$ |
| Group: | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ |
| Conductor: | \(14944\)\(\medspace = 2^{5} \cdot 467 \) |
| Artin stem field: | Galois closure of 9.5.208583809024.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | 12T175 |
| Parity: | even |
| Determinant: | 1.3736.2t1.a.a |
| Projective image: | $C_3^3:S_4$ |
| Projective stem field: | Galois closure of 9.5.208583809024.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 3x^{7} - 5x^{6} + 3x^{5} + 10x^{4} - 7x^{3} - 5x^{2} + 6x + 2 \)
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The roots of $f$ are computed in an extension of $\Q_{ 179 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 179 }$:
\( x^{3} + 4x + 177 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 79 + 96\cdot 179 + 49\cdot 179^{2} + 94\cdot 179^{3} + 5\cdot 179^{4} + 10\cdot 179^{5} + 160\cdot 179^{6} + 149\cdot 179^{7} + 157\cdot 179^{8} + 179^{9} +O(179^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 102 + 127\cdot 179 + 157\cdot 179^{2} + 169\cdot 179^{3} + 21\cdot 179^{4} + 34\cdot 179^{5} + 149\cdot 179^{6} + 21\cdot 179^{7} + 125\cdot 179^{8} + 18\cdot 179^{9} +O(179^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 177 + 133\cdot 179 + 150\cdot 179^{2} + 93\cdot 179^{3} + 151\cdot 179^{4} + 134\cdot 179^{5} + 48\cdot 179^{6} + 7\cdot 179^{7} + 75\cdot 179^{8} + 158\cdot 179^{9} +O(179^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 6 a^{2} + 122 a + 16 + \left(53 a^{2} + 146 a + 22\right)\cdot 179 + \left(155 a^{2} + 176 a + 56\right)\cdot 179^{2} + \left(64 a^{2} + 136 a + 113\right)\cdot 179^{3} + \left(39 a^{2} + 165 a + 164\right)\cdot 179^{4} + \left(154 a^{2} + 174 a + 112\right)\cdot 179^{5} + \left(86 a^{2} + 113 a + 52\right)\cdot 179^{6} + \left(45 a^{2} + 19 a + 121\right)\cdot 179^{7} + \left(151 a^{2} + 173 a + 164\right)\cdot 179^{8} + \left(163 a^{2} + 105 a + 78\right)\cdot 179^{9} +O(179^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 20 a^{2} + 101 a + 113 + \left(169 a^{2} + 167 a + 152\right)\cdot 179 + \left(174 a^{2} + 38 a + 48\right)\cdot 179^{2} + \left(164 a^{2} + 160 a + 22\right)\cdot 179^{3} + \left(28 a^{2} + 82 a + 77\right)\cdot 179^{4} + \left(34 a^{2} + 76 a + 31\right)\cdot 179^{5} + \left(48 a^{2} + 36 a + 9\right)\cdot 179^{6} + \left(17 a^{2} + 14 a + 46\right)\cdot 179^{7} + \left(178 a^{2} + 18 a + 57\right)\cdot 179^{8} + \left(76 a^{2} + 42 a + 26\right)\cdot 179^{9} +O(179^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 61 a^{2} + 13 a + 103 + \left(169 a^{2} + 117 a + 93\right)\cdot 179 + \left(138 a^{2} + 108 a + 12\right)\cdot 179^{2} + \left(141 a^{2} + 43 a + 20\right)\cdot 179^{3} + \left(2 a^{2} + 51 a + 67\right)\cdot 179^{4} + \left(112 a^{2} + 82 a\right)\cdot 179^{5} + \left(30 a^{2} + 148 a + 22\right)\cdot 179^{6} + \left(167 a^{2} + 151 a + 28\right)\cdot 179^{7} + \left(60 a^{2} + 132 a + 43\right)\cdot 179^{8} + \left(142 a^{2} + 120 a + 81\right)\cdot 179^{9} +O(179^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 112 a^{2} + 44 a + 60 + \left(135 a^{2} + 94 a + 63\right)\cdot 179 + \left(63 a^{2} + 72 a + 110\right)\cdot 179^{2} + \left(151 a^{2} + 177 a + 45\right)\cdot 179^{3} + \left(136 a^{2} + 140 a + 126\right)\cdot 179^{4} + \left(91 a^{2} + 100 a + 65\right)\cdot 179^{5} + \left(61 a^{2} + 95 a + 104\right)\cdot 179^{6} + \left(145 a^{2} + 7 a + 29\right)\cdot 179^{7} + \left(145 a^{2} + 52 a + 150\right)\cdot 179^{8} + \left(51 a^{2} + 131 a + 18\right)\cdot 179^{9} +O(179^{10})\)
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| $r_{ 8 }$ | $=$ |
\( 161 a^{2} + 60 a + 131 + \left(113 a^{2} + 93 a + 124\right)\cdot 179 + \left(70 a^{2} + 5 a + 128\right)\cdot 179^{2} + \left(113 a^{2} + 26 a + 63\right)\cdot 179^{3} + \left(119 a^{2} + 30 a + 80\right)\cdot 179^{4} + \left(55 a^{2} + 109 a + 148\right)\cdot 179^{5} + \left(18 a^{2} + 136 a + 48\right)\cdot 179^{6} + \left(115 a^{2} + 75 a + 68\right)\cdot 179^{7} + \left(70 a^{2} + 95 a + 9\right)\cdot 179^{8} + \left(59 a^{2} + 155 a + 39\right)\cdot 179^{9} +O(179^{10})\)
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| $r_{ 9 }$ | $=$ |
\( 177 a^{2} + 18 a + 114 + \left(74 a^{2} + 97 a + 80\right)\cdot 179 + \left(112 a^{2} + 134 a + 1\right)\cdot 179^{2} + \left(79 a^{2} + 171 a + 93\right)\cdot 179^{3} + \left(30 a^{2} + 65 a + 21\right)\cdot 179^{4} + \left(89 a^{2} + 172 a + 178\right)\cdot 179^{5} + \left(112 a^{2} + 5 a + 120\right)\cdot 179^{6} + \left(46 a^{2} + 89 a + 64\right)\cdot 179^{7} + \left(109 a^{2} + 65 a + 112\right)\cdot 179^{8} + \left(42 a^{2} + 160 a + 113\right)\cdot 179^{9} +O(179^{10})\)
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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 |
| $1$ | $1$ | $()$ | $4$ |
| $18$ | $2$ | $(1,4)(2,6)(3,7)$ | $2$ |
| $27$ | $2$ | $(1,3)(4,6)$ | $0$ |
| $4$ | $3$ | $(1,3,2)(4,6,7)(5,8,9)$ | $-3 \zeta_{3} - 2$ |
| $4$ | $3$ | $(1,2,3)(4,7,6)(5,9,8)$ | $3 \zeta_{3} + 1$ |
| $6$ | $3$ | $(1,2,3)$ | $-2$ |
| $12$ | $3$ | $(1,3,2)(4,7,6)$ | $1$ |
| $72$ | $3$ | $(1,9,4)(2,8,7)(3,5,6)$ | $1$ |
| $162$ | $4$ | $(1,6,3,4)(2,7)(8,9)$ | $0$ |
| $18$ | $6$ | $(1,4)(2,6)(3,7)(5,8,9)$ | $-2 \zeta_{3} - 2$ |
| $18$ | $6$ | $(1,4)(2,6)(3,7)(5,9,8)$ | $2 \zeta_{3}$ |
| $36$ | $6$ | $(1,6,3,4,2,7)(5,8,9)$ | $\zeta_{3} + 1$ |
| $36$ | $6$ | $(1,7,2,4,3,6)(5,9,8)$ | $-\zeta_{3}$ |
| $36$ | $6$ | $(1,7,2,4,3,6)$ | $-1$ |
| $54$ | $6$ | $(1,3,2)(4,6)(5,8)$ | $0$ |
| $72$ | $9$ | $(1,8,4,3,9,6,2,5,7)$ | $\zeta_{3}$ |
| $72$ | $9$ | $(1,4,9,2,7,8,3,6,5)$ | $-\zeta_{3} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.