Basic invariants
| Dimension: | $6$ |
| Group: | $C_3^2 : D_{6} $ |
| Conductor: | \(5378240000\)\(\medspace = 2^{9} \cdot 5^{4} \cdot 7^{5} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.1.941192000000.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T51 |
| Parity: | odd |
| Determinant: | 1.56.2t1.b.a |
| Projective image: | $C_3^2:D_6$ |
| Projective stem field: | Galois closure of 9.1.941192000000.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 2x^{8} + 8x^{7} - 21x^{6} + 21x^{5} - 7x^{4} - 7x^{3} + 8x^{2} + 5x + 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 137 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 137 }$:
\( x^{3} + 6x + 134 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 21 + 45\cdot 137 + 115\cdot 137^{2} + 119\cdot 137^{3} + 66\cdot 137^{4} + 132\cdot 137^{5} + 132\cdot 137^{6} + 76\cdot 137^{7} + 77\cdot 137^{8} + 32\cdot 137^{9} +O(137^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 88 + 94\cdot 137 + 11\cdot 137^{2} + 58\cdot 137^{3} + 122\cdot 137^{4} + 108\cdot 137^{5} + 126\cdot 137^{6} + 39\cdot 137^{7} + 119\cdot 137^{8} + 28\cdot 137^{9} +O(137^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 92 + 131\cdot 137 + 72\cdot 137^{2} + 18\cdot 137^{3} + 106\cdot 137^{4} + 9\cdot 137^{5} + 121\cdot 137^{6} + 93\cdot 137^{7} + 87\cdot 137^{8} + 123\cdot 137^{9} +O(137^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 24 a^{2} + 36 a + 85 + \left(48 a^{2} + 19 a + 45\right)\cdot 137 + \left(129 a^{2} + 10 a + 118\right)\cdot 137^{2} + \left(14 a^{2} + 19 a + 51\right)\cdot 137^{3} + \left(34 a^{2} + 103 a + 77\right)\cdot 137^{4} + \left(6 a^{2} + 116 a + 57\right)\cdot 137^{5} + \left(60 a^{2} + 43 a + 30\right)\cdot 137^{6} + \left(105 a^{2} + 96 a + 79\right)\cdot 137^{7} + \left(107 a^{2} + 9 a + 82\right)\cdot 137^{8} + \left(132 a^{2} + 33 a + 35\right)\cdot 137^{9} +O(137^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 54 a^{2} + 120 a + 68 + \left(41 a^{2} + 90 a + 18\right)\cdot 137 + \left(3 a^{2} + 92 a + 25\right)\cdot 137^{2} + \left(83 a^{2} + 56 a + 50\right)\cdot 137^{3} + \left(80 a^{2} + 135 a + 126\right)\cdot 137^{4} + \left(16 a^{2} + 54 a + 98\right)\cdot 137^{5} + \left(70 a^{2} + 48 a + 70\right)\cdot 137^{6} + \left(84 a^{2} + 136 a + 132\right)\cdot 137^{7} + \left(115 a^{2} + 73 a + 113\right)\cdot 137^{8} + \left(54 a^{2} + 13 a + 134\right)\cdot 137^{9} +O(137^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 59 a^{2} + 118 a + 88 + \left(47 a^{2} + 26 a + 42\right)\cdot 137 + \left(4 a^{2} + 34 a + 29\right)\cdot 137^{2} + \left(39 a^{2} + 61 a + 11\right)\cdot 137^{3} + \left(22 a^{2} + 35 a + 30\right)\cdot 137^{4} + \left(114 a^{2} + 102 a + 78\right)\cdot 137^{5} + \left(6 a^{2} + 44 a + 91\right)\cdot 137^{6} + \left(84 a^{2} + 41 a + 130\right)\cdot 137^{7} + \left(50 a^{2} + 53 a + 127\right)\cdot 137^{8} + \left(86 a^{2} + 90 a + 123\right)\cdot 137^{9} +O(137^{10})\)
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| $r_{ 7 }$ | $=$ |
\( 64 a^{2} + 107 a + 18 + \left(108 a^{2} + 64 a + 125\right)\cdot 137 + \left(5 a + 15\right)\cdot 137^{2} + \left(89 a^{2} + 31 a + 70\right)\cdot 137^{3} + \left(77 a^{2} + 117 a + 88\right)\cdot 137^{4} + \left(119 a^{2} + 68 a + 133\right)\cdot 137^{5} + \left(134 a^{2} + 128 a + 119\right)\cdot 137^{6} + \left(105 a^{2} + 36 a + 56\right)\cdot 137^{7} + \left(25 a^{2} + 47 a + 128\right)\cdot 137^{8} + \left(68 a^{2} + 104 a + 112\right)\cdot 137^{9} +O(137^{10})\)
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| $r_{ 8 }$ | $=$ |
\( 93 a^{2} + 105 a + 134 + \left(65 a^{2} + 110 a + 90\right)\cdot 137 + \left(28 a^{2} + 92 a + 126\right)\cdot 137^{2} + \left(57 a^{2} + 25 a + 79\right)\cdot 137^{3} + \left(3 a^{2} + 111 a + 65\right)\cdot 137^{4} + \left(15 a^{2} + 126 a + 126\right)\cdot 137^{5} + \left(23 a^{2} + 81 a + 83\right)\cdot 137^{6} + \left(37 a^{2} + 42 a + 55\right)\cdot 137^{7} + \left(46 a^{2} + 70 a + 73\right)\cdot 137^{8} + \left(27 a^{2} + 60 a + 86\right)\cdot 137^{9} +O(137^{10})\)
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| $r_{ 9 }$ | $=$ |
\( 117 a^{2} + 62 a + 93 + \left(99 a^{2} + 98 a + 90\right)\cdot 137 + \left(107 a^{2} + 38 a + 32\right)\cdot 137^{2} + \left(127 a^{2} + 80 a + 88\right)\cdot 137^{3} + \left(55 a^{2} + 45 a + 1\right)\cdot 137^{4} + \left(2 a^{2} + 78 a + 76\right)\cdot 137^{5} + \left(116 a^{2} + 63 a + 44\right)\cdot 137^{6} + \left(130 a^{2} + 57 a + 19\right)\cdot 137^{7} + \left(64 a^{2} + 19 a + 11\right)\cdot 137^{8} + \left(41 a^{2} + 109 a + 6\right)\cdot 137^{9} +O(137^{10})\)
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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 |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(1,3)(4,5)(7,8)$ | $0$ |
| $9$ | $2$ | $(1,7)(2,9)(3,8)$ | $-2$ |
| $9$ | $2$ | $(1,8)(2,9)(3,7)(4,5)$ | $0$ |
| $2$ | $3$ | $(1,2,3)(4,5,6)(7,9,8)$ | $-3$ |
| $6$ | $3$ | $(1,7,5)(2,9,6)(3,8,4)$ | $0$ |
| $6$ | $3$ | $(4,5,6)(7,8,9)$ | $0$ |
| $12$ | $3$ | $(1,5,8)(2,6,7)(3,4,9)$ | $0$ |
| $18$ | $6$ | $(1,8,5,3,7,4)(2,9,6)$ | $0$ |
| $18$ | $6$ | $(1,9,3,7,2,8)(4,5,6)$ | $1$ |
| $18$ | $6$ | $(2,3)(4,8,5,9,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.