Basic invariants
| Dimension: | $18$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(971\!\cdots\!000\)\(\medspace = 2^{28} \cdot 3^{32} \cdot 5^{9} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.1049760000.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 36T1758 |
| Parity: | odd |
| Determinant: | 1.20.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.2.1049760000.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 3x^{6} - 2x^{5} + 12x^{3} - 13x^{2} + 6x + 9 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 131 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 131 }$:
\( x^{2} + 127x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 106 a + 35 + \left(96 a + 5\right)\cdot 131 + \left(109 a + 113\right)\cdot 131^{2} + \left(92 a + 80\right)\cdot 131^{3} + \left(93 a + 22\right)\cdot 131^{4} + \left(11 a + 23\right)\cdot 131^{5} + \left(127 a + 112\right)\cdot 131^{6} + \left(26 a + 15\right)\cdot 131^{7} + \left(26 a + 129\right)\cdot 131^{8} + \left(50 a + 103\right)\cdot 131^{9} +O(131^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 106 + 36\cdot 131 + 122\cdot 131^{2} + 8\cdot 131^{3} + 114\cdot 131^{4} + 131^{5} + 115\cdot 131^{6} + 122\cdot 131^{7} + 62\cdot 131^{8} + 56\cdot 131^{9} +O(131^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 128 a + 77 + \left(46 a + 113\right)\cdot 131 + \left(103 a + 82\right)\cdot 131^{2} + \left(47 a + 115\right)\cdot 131^{3} + \left(128 a + 89\right)\cdot 131^{4} + \left(87 a + 4\right)\cdot 131^{5} + \left(4 a + 23\right)\cdot 131^{6} + \left(22 a + 32\right)\cdot 131^{7} + \left(21 a + 81\right)\cdot 131^{8} + \left(63 a + 56\right)\cdot 131^{9} +O(131^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 7 a + 1 + \left(44 a + 97\right)\cdot 131 + \left(4 a + 56\right)\cdot 131^{2} + \left(44 a + 95\right)\cdot 131^{3} + \left(16 a + 87\right)\cdot 131^{4} + \left(31 a + 11\right)\cdot 131^{5} + \left(17 a + 79\right)\cdot 131^{6} + \left(59 a + 80\right)\cdot 131^{7} + \left(59 a + 4\right)\cdot 131^{8} + \left(15 a + 4\right)\cdot 131^{9} +O(131^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 14 + 69\cdot 131 + 65\cdot 131^{3} + 26\cdot 131^{4} + 27\cdot 131^{5} + 39\cdot 131^{6} + 122\cdot 131^{7} + 104\cdot 131^{8} + 122\cdot 131^{9} +O(131^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 124 a + 29 + \left(86 a + 4\right)\cdot 131 + \left(126 a + 30\right)\cdot 131^{2} + \left(86 a + 5\right)\cdot 131^{3} + \left(114 a + 109\right)\cdot 131^{4} + \left(99 a + 119\right)\cdot 131^{5} + \left(113 a + 116\right)\cdot 131^{6} + \left(71 a + 37\right)\cdot 131^{7} + \left(71 a + 52\right)\cdot 131^{8} + \left(115 a + 6\right)\cdot 131^{9} +O(131^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 3 a + 65 + \left(84 a + 42\right)\cdot 131 + \left(27 a + 56\right)\cdot 131^{2} + \left(83 a + 72\right)\cdot 131^{3} + \left(2 a + 31\right)\cdot 131^{4} + \left(43 a + 97\right)\cdot 131^{5} + \left(126 a + 84\right)\cdot 131^{6} + \left(108 a + 115\right)\cdot 131^{7} + \left(109 a + 12\right)\cdot 131^{8} + \left(67 a + 26\right)\cdot 131^{9} +O(131^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 25 a + 66 + \left(34 a + 24\right)\cdot 131 + \left(21 a + 62\right)\cdot 131^{2} + \left(38 a + 80\right)\cdot 131^{3} + \left(37 a + 42\right)\cdot 131^{4} + \left(119 a + 107\right)\cdot 131^{5} + \left(3 a + 84\right)\cdot 131^{6} + \left(104 a + 127\right)\cdot 131^{7} + \left(104 a + 75\right)\cdot 131^{8} + \left(80 a + 16\right)\cdot 131^{9} +O(131^{10})\)
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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 |
| $1$ | $1$ | $()$ | $18$ |
| $6$ | $2$ | $(2,5)(3,7)$ | $-6$ |
| $9$ | $2$ | $(1,6)(2,5)(3,7)(4,8)$ | $2$ |
| $12$ | $2$ | $(1,4)$ | $0$ |
| $24$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $36$ | $2$ | $(1,4)(2,3)$ | $-2$ |
| $36$ | $2$ | $(1,4)(2,5)(3,7)$ | $0$ |
| $16$ | $3$ | $(1,6,8)$ | $0$ |
| $64$ | $3$ | $(1,6,8)(3,5,7)$ | $0$ |
| $12$ | $4$ | $(2,3,5,7)$ | $0$ |
| $36$ | $4$ | $(1,4,6,8)(2,3,5,7)$ | $-2$ |
| $36$ | $4$ | $(1,4,6,8)(2,5)(3,7)$ | $0$ |
| $72$ | $4$ | $(1,2,6,5)(3,8,7,4)$ | $0$ |
| $72$ | $4$ | $(1,4)(2,3,5,7)$ | $2$ |
| $144$ | $4$ | $(1,3,4,2)(5,6)(7,8)$ | $0$ |
| $48$ | $6$ | $(1,8,6)(2,5)(3,7)$ | $0$ |
| $96$ | $6$ | $(1,4)(3,7,5)$ | $0$ |
| $192$ | $6$ | $(1,3,6,5,8,7)(2,4)$ | $0$ |
| $144$ | $8$ | $(1,2,4,3,6,5,8,7)$ | $0$ |
| $96$ | $12$ | $(1,6,8)(2,3,5,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.