Basic invariants
| Dimension: | $9$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(107\!\cdots\!576\)\(\medspace = 2^{12} \cdot 13^{7} \cdot 53^{5} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.4.14150908918016.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T274 |
| Parity: | even |
| Determinant: | 1.689.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.4.14150908918016.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 8x^{6} + 6x^{5} - 229x^{4} + 750x^{3} - 530x^{2} - 70x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 103 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 103 }$:
\( x^{2} + 102x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 101 a + 87 + \left(39 a + 34\right)\cdot 103 + \left(47 a + 31\right)\cdot 103^{2} + \left(91 a + 27\right)\cdot 103^{3} + \left(78 a + 65\right)\cdot 103^{4} + \left(55 a + 19\right)\cdot 103^{5} + \left(59 a + 75\right)\cdot 103^{6} + \left(68 a + 32\right)\cdot 103^{7} + \left(93 a + 20\right)\cdot 103^{8} + \left(35 a + 46\right)\cdot 103^{9} +O(103^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 45 a + 47 + 35 a\cdot 103 + \left(25 a + 73\right)\cdot 103^{2} + \left(102 a + 66\right)\cdot 103^{3} + \left(84 a + 52\right)\cdot 103^{4} + \left(41 a + 13\right)\cdot 103^{5} + \left(79 a + 7\right)\cdot 103^{6} + \left(a + 53\right)\cdot 103^{7} + \left(72 a + 86\right)\cdot 103^{8} + \left(25 a + 5\right)\cdot 103^{9} +O(103^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 27 a + 31 + \left(62 a + 11\right)\cdot 103 + \left(30 a + 12\right)\cdot 103^{2} + \left(50 a + 82\right)\cdot 103^{3} + \left(8 a + 29\right)\cdot 103^{4} + \left(76 a + 17\right)\cdot 103^{5} + \left(35 a + 4\right)\cdot 103^{6} + \left(33 a + 89\right)\cdot 103^{7} + \left(81 a + 25\right)\cdot 103^{8} + \left(49 a + 8\right)\cdot 103^{9} +O(103^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 21 + 4\cdot 103 + 8\cdot 103^{2} + 57\cdot 103^{3} + 61\cdot 103^{4} + 20\cdot 103^{5} + 72\cdot 103^{6} + 21\cdot 103^{7} + 95\cdot 103^{8} + 81\cdot 103^{9} +O(103^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 2 a + 85 + \left(63 a + 76\right)\cdot 103 + \left(55 a + 38\right)\cdot 103^{2} + \left(11 a + 71\right)\cdot 103^{3} + \left(24 a + 52\right)\cdot 103^{4} + \left(47 a + 99\right)\cdot 103^{5} + \left(43 a + 78\right)\cdot 103^{6} + \left(34 a + 41\right)\cdot 103^{7} + \left(9 a + 45\right)\cdot 103^{8} + \left(67 a + 91\right)\cdot 103^{9} +O(103^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 58 a + 92 + \left(67 a + 93\right)\cdot 103 + \left(77 a + 62\right)\cdot 103^{2} + 40\cdot 103^{3} + \left(18 a + 35\right)\cdot 103^{4} + \left(61 a + 73\right)\cdot 103^{5} + \left(23 a + 44\right)\cdot 103^{6} + \left(101 a + 78\right)\cdot 103^{7} + \left(30 a + 53\right)\cdot 103^{8} + \left(77 a + 62\right)\cdot 103^{9} +O(103^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 98 + 40\cdot 103 + 102\cdot 103^{2} + 67\cdot 103^{3} + 23\cdot 103^{4} + 83\cdot 103^{5} + 62\cdot 103^{6} + 8\cdot 103^{7} + 11\cdot 103^{8} + 36\cdot 103^{9} +O(103^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 76 a + 58 + \left(40 a + 46\right)\cdot 103 + \left(72 a + 83\right)\cdot 103^{2} + \left(52 a + 101\right)\cdot 103^{3} + \left(94 a + 90\right)\cdot 103^{4} + \left(26 a + 84\right)\cdot 103^{5} + \left(67 a + 66\right)\cdot 103^{6} + \left(69 a + 86\right)\cdot 103^{7} + \left(21 a + 73\right)\cdot 103^{8} + \left(53 a + 79\right)\cdot 103^{9} +O(103^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $9$ |
| $6$ | $2$ | $(1,5)(2,6)$ | $-3$ |
| $9$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $1$ |
| $12$ | $2$ | $(3,4)$ | $3$ |
| $24$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $-3$ |
| $36$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $36$ | $2$ | $(1,5)(2,6)(3,4)$ | $-1$ |
| $16$ | $3$ | $(3,7,8)$ | $0$ |
| $64$ | $3$ | $(2,5,6)(3,7,8)$ | $0$ |
| $12$ | $4$ | $(1,2,5,6)$ | $-3$ |
| $36$ | $4$ | $(1,2,5,6)(3,4,7,8)$ | $1$ |
| $36$ | $4$ | $(1,5)(2,6)(3,4,7,8)$ | $1$ |
| $72$ | $4$ | $(1,7,5,3)(2,8,6,4)$ | $1$ |
| $72$ | $4$ | $(1,2,5,6)(3,4)$ | $-1$ |
| $144$ | $4$ | $(1,3,2,4)(5,7)(6,8)$ | $-1$ |
| $48$ | $6$ | $(1,5)(2,6)(3,8,7)$ | $0$ |
| $96$ | $6$ | $(2,6,5)(3,4)$ | $0$ |
| $192$ | $6$ | $(1,4)(2,7,5,8,6,3)$ | $0$ |
| $144$ | $8$ | $(1,4,2,7,5,8,6,3)$ | $1$ |
| $96$ | $12$ | $(1,2,5,6)(3,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.