Basic invariants
| Dimension: | $12$ |
| Group: | $S_3\wr S_3$ |
| Conductor: | \(36922207099409344\)\(\medspace = 2^{6} \cdot 31^{5} \cdot 67^{4}\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.1.127743808.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T315 |
| Parity: | odd |
| Determinant: | 1.31.2t1.a.a |
| Projective image: | $C_3^3.S_4.C_2$ |
| Projective stem field: | Galois closure of 9.1.127743808.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 2x^{8} + 3x^{7} - 3x^{6} + x^{5} + 4x^{4} - 8x^{3} + 9x^{2} - 5x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 149 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 149 }$:
\( x^{3} + 3x + 147 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 33 + 26\cdot 149 + 103\cdot 149^{2} + 93\cdot 149^{3} + 50\cdot 149^{4} + 30\cdot 149^{5} + 89\cdot 149^{6} + 18\cdot 149^{7} + 111\cdot 149^{8} + 129\cdot 149^{9} +O(149^{10})\)
| $r_{ 2 }$ |
$=$ |
\( 53 + 81\cdot 149 + 101\cdot 149^{2} + 143\cdot 149^{3} + 74\cdot 149^{4} + 7\cdot 149^{5} + 132\cdot 149^{6} + 139\cdot 149^{7} + 146\cdot 149^{8} + 98\cdot 149^{9} +O(149^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 91 + 43\cdot 149 + 58\cdot 149^{2} + 23\cdot 149^{3} + 107\cdot 149^{4} + 110\cdot 149^{5} + 60\cdot 149^{6} + 72\cdot 149^{7} + 91\cdot 149^{8} + 31\cdot 149^{9} +O(149^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 7 a^{2} + 82 a + 107 + \left(99 a^{2} + 12 a + 9\right)\cdot 149 + \left(39 a^{2} + 14 a + 140\right)\cdot 149^{2} + \left(106 a^{2} + 46 a + 134\right)\cdot 149^{3} + \left(43 a^{2} + 126 a + 15\right)\cdot 149^{4} + \left(58 a^{2} + 8 a + 22\right)\cdot 149^{5} + \left(117 a + 118\right)\cdot 149^{6} + \left(132 a^{2} + 23 a + 51\right)\cdot 149^{7} + \left(51 a^{2} + 86 a + 77\right)\cdot 149^{8} + \left(94 a^{2} + 33 a + 105\right)\cdot 149^{9} +O(149^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 55 a^{2} + 91 a + 54 + \left(121 a^{2} + 20 a + 54\right)\cdot 149 + \left(117 a^{2} + 61 a + 147\right)\cdot 149^{2} + \left(137 a^{2} + 120 a + 48\right)\cdot 149^{3} + \left(13 a^{2} + 127 a + 105\right)\cdot 149^{4} + \left(49 a^{2} + 107 a + 3\right)\cdot 149^{5} + \left(87 a^{2} + 19 a + 143\right)\cdot 149^{6} + \left(64 a^{2} + 8 a + 65\right)\cdot 149^{7} + \left(77 a^{2} + 83 a + 128\right)\cdot 149^{8} + \left(66 a^{2} + 29 a + 49\right)\cdot 149^{9} +O(149^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 82 a^{2} + 105 a + 112 + \left(33 a^{2} + 118 a + 105\right)\cdot 149 + \left(123 a^{2} + 6 a + 147\right)\cdot 149^{2} + \left(22 a^{2} + 55 a + 85\right)\cdot 149^{3} + \left(39 a^{2} + 117 a + 22\right)\cdot 149^{4} + \left(139 a^{2} + 120 a + 75\right)\cdot 149^{5} + \left(40 a^{2} + 117 a + 19\right)\cdot 149^{6} + \left(19 a^{2} + 29 a + 124\right)\cdot 149^{7} + \left(93 a^{2} + 33 a + 145\right)\cdot 149^{8} + \left(70 a^{2} + 73 a + 87\right)\cdot 149^{9} +O(149^{10})\)
| $r_{ 7 }$ |
$=$ |
\( 87 a^{2} + 125 a + 118 + \left(77 a^{2} + 115 a + 115\right)\cdot 149 + \left(140 a^{2} + 73 a + 43\right)\cdot 149^{2} + \left(53 a^{2} + 131 a + 30\right)\cdot 149^{3} + \left(91 a^{2} + 43 a + 111\right)\cdot 149^{4} + \left(41 a^{2} + 32 a + 137\right)\cdot 149^{5} + \left(61 a^{2} + 12 a + 90\right)\cdot 149^{6} + \left(101 a^{2} + 117 a + 139\right)\cdot 149^{7} + \left(19 a^{2} + 128 a + 12\right)\cdot 149^{8} + \left(137 a^{2} + 85 a + 42\right)\cdot 149^{9} +O(149^{10})\)
| $r_{ 8 }$ |
$=$ |
\( 99 a^{2} + 46 a + 146 + \left(98 a^{2} + 6 a + 86\right)\cdot 149 + \left(7 a^{2} + 48 a + 65\right)\cdot 149^{2} + \left(124 a^{2} + 101 a + 139\right)\cdot 149^{3} + \left(114 a^{2} + 58 a + 24\right)\cdot 149^{4} + \left(66 a^{2} + 111 a + 79\right)\cdot 149^{5} + \left(108 a^{2} + 31 a + 5\right)\cdot 149^{6} + \left(41 a^{2} + 5 a + 20\right)\cdot 149^{7} + \left(23 a^{2} + 120 a + 6\right)\cdot 149^{8} + \left(129 a^{2} + 133 a + 56\right)\cdot 149^{9} +O(149^{10})\)
| $r_{ 9 }$ |
$=$ |
\( 117 a^{2} + 147 a + 33 + \left(16 a^{2} + 23 a + 72\right)\cdot 149 + \left(18 a^{2} + 94 a + 86\right)\cdot 149^{2} + \left(2 a^{2} + 141 a + 44\right)\cdot 149^{3} + \left(144 a^{2} + 121 a + 83\right)\cdot 149^{4} + \left(91 a^{2} + 65 a + 129\right)\cdot 149^{5} + \left(148 a^{2} + 148 a + 85\right)\cdot 149^{6} + \left(87 a^{2} + 113 a + 112\right)\cdot 149^{7} + \left(32 a^{2} + 144 a + 24\right)\cdot 149^{8} + \left(98 a^{2} + 90 a + 143\right)\cdot 149^{9} +O(149^{10})\)
| |
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 value |
| $1$ | $1$ | $()$ | $12$ |
| $9$ | $2$ | $(6,8)$ | $4$ |
| $18$ | $2$ | $(1,6)(2,8)(3,9)$ | $2$ |
| $27$ | $2$ | $(1,2)(4,5)(6,8)$ | $0$ |
| $27$ | $2$ | $(1,2)(6,8)$ | $0$ |
| $54$ | $2$ | $(1,4)(2,5)(3,7)(6,8)$ | $2$ |
| $6$ | $3$ | $(4,5,7)$ | $0$ |
| $8$ | $3$ | $(1,2,3)(4,5,7)(6,8,9)$ | $3$ |
| $12$ | $3$ | $(1,2,3)(4,5,7)$ | $-3$ |
| $72$ | $3$ | $(1,4,6)(2,5,8)(3,7,9)$ | $0$ |
| $54$ | $4$ | $(1,6,2,8)(3,9)$ | $0$ |
| $162$ | $4$ | $(2,3)(4,6,5,8)(7,9)$ | $0$ |
| $36$ | $6$ | $(1,6)(2,8)(3,9)(4,5,7)$ | $2$ |
| $36$ | $6$ | $(4,8,5,9,7,6)$ | $-1$ |
| $36$ | $6$ | $(4,5,7)(6,8)$ | $-2$ |
| $36$ | $6$ | $(1,2,3)(4,5,7)(6,8)$ | $1$ |
| $54$ | $6$ | $(1,2)(4,7,5)(6,8)$ | $0$ |
| $72$ | $6$ | $(1,8,2,9,3,6)(4,5,7)$ | $-1$ |
| $108$ | $6$ | $(1,4,2,5,3,7)(6,8)$ | $-1$ |
| $216$ | $6$ | $(1,4,6,2,5,8)(3,7,9)$ | $0$ |
| $144$ | $9$ | $(1,4,8,2,5,9,3,7,6)$ | $0$ |
| $108$ | $12$ | $(1,6,2,8)(3,9)(4,5,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.