Basic invariants
| Dimension: | $18$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(115\!\cdots\!288\)\(\medspace = 2^{65} \cdot 3^{22} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.4.587068342272.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 36T1758 |
| Parity: | even |
| Determinant: | 1.8.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.4.587068342272.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 16x^{5} + 6x^{4} - 32x^{2} + 3 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 149 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 149 }$:
\( x^{2} + 145x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 89 a + 52 + \left(21 a + 20\right)\cdot 149 + \left(27 a + 103\right)\cdot 149^{2} + \left(81 a + 29\right)\cdot 149^{3} + \left(35 a + 45\right)\cdot 149^{4} + \left(74 a + 32\right)\cdot 149^{5} + \left(125 a + 4\right)\cdot 149^{6} + \left(110 a + 55\right)\cdot 149^{7} + \left(139 a + 144\right)\cdot 149^{8} + 81\cdot 149^{9} +O(149^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 29 a + 12 + \left(108 a + 46\right)\cdot 149 + \left(26 a + 76\right)\cdot 149^{2} + \left(111 a + 147\right)\cdot 149^{3} + \left(137 a + 39\right)\cdot 149^{4} + \left(15 a + 93\right)\cdot 149^{5} + \left(31 a + 119\right)\cdot 149^{6} + \left(143 a + 10\right)\cdot 149^{7} + \left(39 a + 39\right)\cdot 149^{8} + \left(11 a + 24\right)\cdot 149^{9} +O(149^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 120 a + 128 + \left(40 a + 2\right)\cdot 149 + \left(122 a + 75\right)\cdot 149^{2} + \left(37 a + 118\right)\cdot 149^{3} + \left(11 a + 32\right)\cdot 149^{4} + \left(133 a + 19\right)\cdot 149^{5} + \left(117 a + 79\right)\cdot 149^{6} + \left(5 a + 105\right)\cdot 149^{7} + \left(109 a + 55\right)\cdot 149^{8} + \left(137 a + 29\right)\cdot 149^{9} +O(149^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 57 a + 103 + \left(64 a + 29\right)\cdot 149 + \left(a + 106\right)\cdot 149^{2} + \left(52 a + 90\right)\cdot 149^{3} + \left(114 a + 19\right)\cdot 149^{4} + \left(146 a + 122\right)\cdot 149^{5} + \left(113 a + 148\right)\cdot 149^{6} + 32 a\cdot 149^{7} + \left(36 a + 97\right)\cdot 149^{8} + 76 a\cdot 149^{9} +O(149^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 92 a + 33 + \left(84 a + 81\right)\cdot 149 + \left(147 a + 47\right)\cdot 149^{2} + \left(96 a + 148\right)\cdot 149^{3} + \left(34 a + 126\right)\cdot 149^{4} + \left(2 a + 147\right)\cdot 149^{5} + \left(35 a + 10\right)\cdot 149^{6} + \left(116 a + 18\right)\cdot 149^{7} + \left(112 a + 60\right)\cdot 149^{8} + \left(72 a + 120\right)\cdot 149^{9} +O(149^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 96 + 109\cdot 149 + 91\cdot 149^{2} + 60\cdot 149^{3} + 144\cdot 149^{4} + 44\cdot 149^{5} + 41\cdot 149^{6} + 75\cdot 149^{7} + 59\cdot 149^{8} + 30\cdot 149^{9} +O(149^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 62 + 139\cdot 149 + 54\cdot 149^{2} + 120\cdot 149^{3} + 80\cdot 149^{4} + 140\cdot 149^{5} + 57\cdot 149^{6} + 106\cdot 149^{7} + 143\cdot 149^{8} + 64\cdot 149^{9} +O(149^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 60 a + 110 + \left(127 a + 17\right)\cdot 149 + \left(121 a + 41\right)\cdot 149^{2} + \left(67 a + 29\right)\cdot 149^{3} + \left(113 a + 106\right)\cdot 149^{4} + \left(74 a + 144\right)\cdot 149^{5} + \left(23 a + 133\right)\cdot 149^{6} + \left(38 a + 74\right)\cdot 149^{7} + \left(9 a + 145\right)\cdot 149^{8} + \left(148 a + 94\right)\cdot 149^{9} +O(149^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $18$ |
| $6$ | $2$ | $(2,6)(3,7)$ | $-6$ |
| $9$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $2$ |
| $12$ | $2$ | $(1,4)$ | $0$ |
| $24$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $36$ | $2$ | $(1,4)(2,3)$ | $-2$ |
| $36$ | $2$ | $(1,4)(2,6)(3,7)$ | $0$ |
| $16$ | $3$ | $(1,5,8)$ | $0$ |
| $64$ | $3$ | $(1,5,8)(3,6,7)$ | $0$ |
| $12$ | $4$ | $(2,3,6,7)$ | $0$ |
| $36$ | $4$ | $(1,4,5,8)(2,3,6,7)$ | $-2$ |
| $36$ | $4$ | $(1,4,5,8)(2,6)(3,7)$ | $0$ |
| $72$ | $4$ | $(1,2,5,6)(3,8,7,4)$ | $0$ |
| $72$ | $4$ | $(1,4)(2,3,6,7)$ | $2$ |
| $144$ | $4$ | $(1,3,4,2)(5,6)(7,8)$ | $0$ |
| $48$ | $6$ | $(1,8,5)(2,6)(3,7)$ | $0$ |
| $96$ | $6$ | $(1,4)(3,7,6)$ | $0$ |
| $192$ | $6$ | $(1,3,5,6,8,7)(2,4)$ | $0$ |
| $144$ | $8$ | $(1,2,4,3,5,6,8,7)$ | $0$ |
| $96$ | $12$ | $(1,5,8)(2,3,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.