Basic invariants
| Dimension: | $9$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(100\!\cdots\!125\)\(\medspace = 5^{7} \cdot 13^{7} \cdot 29^{5} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.4.28298328053125.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T274 |
| Parity: | even |
| Determinant: | 1.1885.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.4.28298328053125.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 23x^{6} - 12x^{5} + 229x^{4} + 73x^{3} - 971x^{2} + 37x + 1226 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 139 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 139 }$:
\( x^{3} + 6x + 137 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 77 + 50\cdot 139 + 133\cdot 139^{2} + 26\cdot 139^{3} + 98\cdot 139^{4} + 85\cdot 139^{5} + 105\cdot 139^{6} + 60\cdot 139^{7} + 58\cdot 139^{8} + 27\cdot 139^{9} +O(139^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 83 + 92\cdot 139 + 84\cdot 139^{2} + 11\cdot 139^{3} + 118\cdot 139^{4} + 66\cdot 139^{5} + 41\cdot 139^{6} + 68\cdot 139^{7} + 52\cdot 139^{8} + 138\cdot 139^{9} +O(139^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 4 a^{2} + 9 a + 81 + \left(119 a^{2} + 59 a + 74\right)\cdot 139 + \left(104 a^{2} + 4 a + 20\right)\cdot 139^{2} + \left(77 a^{2} + 102 a + 29\right)\cdot 139^{3} + \left(21 a^{2} + 116 a + 93\right)\cdot 139^{4} + \left(87 a^{2} + 57 a + 94\right)\cdot 139^{5} + \left(35 a^{2} + 48 a + 128\right)\cdot 139^{6} + \left(79 a^{2} + 73 a + 108\right)\cdot 139^{7} + \left(40 a^{2} + 61 a + 5\right)\cdot 139^{8} + \left(20 a^{2} + 9 a + 35\right)\cdot 139^{9} +O(139^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 11 a^{2} + 118 a + 109 + \left(108 a^{2} + 23 a + 30\right)\cdot 139 + \left(137 a^{2} + 4 a + 13\right)\cdot 139^{2} + \left(75 a^{2} + 76 a + 22\right)\cdot 139^{3} + \left(5 a^{2} + 82 a + 29\right)\cdot 139^{4} + \left(113 a^{2} + 108 a + 59\right)\cdot 139^{5} + \left(120 a^{2} + 38 a + 52\right)\cdot 139^{6} + \left(126 a^{2} + 69 a + 21\right)\cdot 139^{7} + \left(76 a^{2} + 69 a + 12\right)\cdot 139^{8} + \left(128 a^{2} + 125 a + 51\right)\cdot 139^{9} +O(139^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 34 a^{2} + 13 a + 64 + \left(89 a^{2} + 60 a + 108\right)\cdot 139 + \left(25 a^{2} + 81 a + 11\right)\cdot 139^{2} + \left(48 a^{2} + 102 a + 91\right)\cdot 139^{3} + \left(122 a^{2} + 109 a + 39\right)\cdot 139^{4} + \left(10 a^{2} + 20 a + 61\right)\cdot 139^{5} + \left(16 a^{2} + 120 a + 75\right)\cdot 139^{6} + \left(71 a^{2} + 5 a + 32\right)\cdot 139^{7} + \left(62 a^{2} + 114 a + 45\right)\cdot 139^{8} + \left(9 a^{2} + 97 a + 121\right)\cdot 139^{9} +O(139^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 50 a^{2} + 14 a + 128 + \left(61 a^{2} + 93 a + 135\right)\cdot 139 + \left(130 a^{2} + 110 a + 13\right)\cdot 139^{2} + \left(43 a^{2} + 63 a + 74\right)\cdot 139^{3} + \left(27 a^{2} + 32 a + 76\right)\cdot 139^{4} + \left(109 a^{2} + 15 a + 37\right)\cdot 139^{5} + \left(122 a^{2} + 53 a + 85\right)\cdot 139^{6} + \left(59 a^{2} + 29 a + 126\right)\cdot 139^{7} + \left(130 a^{2} + 138 a + 38\right)\cdot 139^{8} + \left(83 a^{2} + 11 a + 2\right)\cdot 139^{9} +O(139^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 55 a^{2} + 112 a + 9 + \left(127 a^{2} + 124 a + 122\right)\cdot 139 + \left(121 a^{2} + 85 a + 118\right)\cdot 139^{2} + \left(46 a^{2} + 111 a + 85\right)\cdot 139^{3} + \left(128 a^{2} + 135 a + 63\right)\cdot 139^{4} + \left(18 a^{2} + 102 a + 93\right)\cdot 139^{5} + \left(104 a + 11\right)\cdot 139^{6} + \left(8 a^{2} + 103 a + 58\right)\cdot 139^{7} + \left(85 a^{2} + 25 a + 135\right)\cdot 139^{8} + \left(45 a^{2} + 29 a + 126\right)\cdot 139^{9} +O(139^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 124 a^{2} + 12 a + 5 + \left(50 a^{2} + 56 a + 80\right)\cdot 139 + \left(35 a^{2} + 130 a + 20\right)\cdot 139^{2} + \left(124 a^{2} + 99 a + 76\right)\cdot 139^{3} + \left(111 a^{2} + 78 a + 37\right)\cdot 139^{4} + \left(77 a^{2} + 111 a + 57\right)\cdot 139^{5} + \left(121 a^{2} + 51 a + 55\right)\cdot 139^{6} + \left(71 a^{2} + 135 a + 79\right)\cdot 139^{7} + \left(21 a^{2} + 7 a + 68\right)\cdot 139^{8} + \left(129 a^{2} + 4 a + 53\right)\cdot 139^{9} +O(139^{10})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 8 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 8 }$ | Character value |
| $1$ | $1$ | $()$ | $9$ |
| $6$ | $2$ | $(1,6)(5,7)$ | $-3$ |
| $9$ | $2$ | $(1,6)(2,4)(3,8)(5,7)$ | $1$ |
| $12$ | $2$ | $(2,3)$ | $3$ |
| $24$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $-3$ |
| $36$ | $2$ | $(1,5)(2,3)$ | $1$ |
| $36$ | $2$ | $(1,6)(2,3)(5,7)$ | $-1$ |
| $16$ | $3$ | $(2,4,8)$ | $0$ |
| $64$ | $3$ | $(2,4,8)(5,6,7)$ | $0$ |
| $12$ | $4$ | $(1,5,6,7)$ | $-3$ |
| $36$ | $4$ | $(1,5,6,7)(2,3,4,8)$ | $1$ |
| $36$ | $4$ | $(1,6)(2,3,4,8)(5,7)$ | $1$ |
| $72$ | $4$ | $(1,4,6,2)(3,5,8,7)$ | $1$ |
| $72$ | $4$ | $(1,5,6,7)(2,3)$ | $-1$ |
| $144$ | $4$ | $(1,2,5,3)(4,6)(7,8)$ | $-1$ |
| $48$ | $6$ | $(1,6)(2,8,4)(5,7)$ | $0$ |
| $96$ | $6$ | $(2,3)(5,7,6)$ | $0$ |
| $192$ | $6$ | $(1,3)(2,5,4,6,8,7)$ | $0$ |
| $144$ | $8$ | $(1,3,5,4,6,8,7,2)$ | $1$ |
| $96$ | $12$ | $(1,5,6,7)(2,4,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.