Basic invariants
| Dimension: | $12$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(4672845941058572032\)\(\medspace = 2^{8} \cdot 7^{7} \cdot 53^{6} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.43302959728.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 36T1763 |
| Parity: | odd |
| Determinant: | 1.7.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.2.43302959728.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} + 9x^{6} - 32x^{5} + 54x^{4} - 26x^{3} - 12x^{2} - 4x + 4 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 113 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 113 }$:
\( x^{3} + 8x + 110 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 9 + 92\cdot 113 + 21\cdot 113^{2} + 42\cdot 113^{3} + 58\cdot 113^{4} + 76\cdot 113^{5} + 45\cdot 113^{6} + 30\cdot 113^{7} + 39\cdot 113^{8} + 61\cdot 113^{9} +O(113^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 40 + 47\cdot 113 + 75\cdot 113^{2} + 5\cdot 113^{3} + 109\cdot 113^{4} + 11\cdot 113^{5} + 27\cdot 113^{6} + 19\cdot 113^{7} + 58\cdot 113^{8} + 14\cdot 113^{9} +O(113^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 39 a^{2} + 53 a + 95 + \left(16 a^{2} + 50 a + 53\right)\cdot 113 + \left(68 a^{2} + 31 a + 107\right)\cdot 113^{2} + \left(98 a^{2} + 2 a + 65\right)\cdot 113^{3} + \left(32 a^{2} + 56 a + 94\right)\cdot 113^{4} + \left(73 a^{2} + 25 a + 41\right)\cdot 113^{5} + \left(59 a^{2} + 109 a + 72\right)\cdot 113^{6} + \left(23 a^{2} + 4 a + 91\right)\cdot 113^{7} + \left(27 a^{2} + 85 a + 42\right)\cdot 113^{8} + \left(13 a^{2} + 34 a + 98\right)\cdot 113^{9} +O(113^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 50 a^{2} + 82 a + 63 + \left(30 a^{2} + 2 a + 111\right)\cdot 113 + \left(97 a^{2} + 88 a + 101\right)\cdot 113^{2} + \left(66 a^{2} + 77 a + 84\right)\cdot 113^{3} + \left(111 a^{2} + 12 a + 92\right)\cdot 113^{4} + \left(71 a^{2} + 34 a + 100\right)\cdot 113^{5} + \left(36 a^{2} + 15 a + 77\right)\cdot 113^{6} + \left(53 a^{2} + 39 a + 75\right)\cdot 113^{7} + \left(8 a^{2} + 55 a + 39\right)\cdot 113^{8} + \left(81 a^{2} + 35 a + 40\right)\cdot 113^{9} +O(113^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 79 a^{2} + 47 a + 7 + \left(12 a^{2} + 91 a + 72\right)\cdot 113 + \left(4 a^{2} + 26 a + 29\right)\cdot 113^{2} + \left(14 a^{2} + 45 a + 29\right)\cdot 113^{3} + \left(105 a^{2} + 34 a + 103\right)\cdot 113^{4} + \left(55 a^{2} + 68 a + 99\right)\cdot 113^{5} + \left(71 a^{2} + 104 a + 97\right)\cdot 113^{6} + \left(6 a^{2} + 97 a + 76\right)\cdot 113^{7} + \left(54 a^{2} + 52 a + 110\right)\cdot 113^{8} + \left(61 a^{2} + 34 a + 91\right)\cdot 113^{9} +O(113^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 82 a^{2} + 84 a + 83 + \left(a^{2} + 25 a + 33\right)\cdot 113 + \left(82 a^{2} + 60 a + 58\right)\cdot 113^{2} + \left(71 a^{2} + 5 a + 35\right)\cdot 113^{3} + \left(30 a^{2} + 64 a + 75\right)\cdot 113^{4} + \left(80 a^{2} + 6 a + 69\right)\cdot 113^{5} + \left(25 a^{2} + 68 a + 19\right)\cdot 113^{6} + \left(a^{2} + 64 a + 99\right)\cdot 113^{7} + \left(97 a^{2} + 98 a + 59\right)\cdot 113^{8} + \left(2 a^{2} + 26 a + 37\right)\cdot 113^{9} +O(113^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 94 a^{2} + 60 a + 34 + \left(80 a^{2} + 84 a + 41\right)\cdot 113 + \left(46 a^{2} + 77 a + 58\right)\cdot 113^{2} + \left(87 a^{2} + 29 a + 81\right)\cdot 113^{3} + \left(83 a^{2} + 36 a + 19\right)\cdot 113^{4} + \left(73 a^{2} + 72 a + 110\right)\cdot 113^{5} + \left(50 a^{2} + 29 a + 1\right)\cdot 113^{6} + \left(58 a^{2} + 9 a + 103\right)\cdot 113^{7} + \left(7 a^{2} + 72 a + 109\right)\cdot 113^{8} + \left(29 a^{2} + 50 a + 101\right)\cdot 113^{9} +O(113^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 108 a^{2} + 13 a + 11 + \left(83 a^{2} + 84 a\right)\cdot 113 + \left(40 a^{2} + 54 a + 112\right)\cdot 113^{2} + \left(65 a + 106\right)\cdot 113^{3} + \left(88 a^{2} + 22 a + 11\right)\cdot 113^{4} + \left(96 a^{2} + 19 a + 54\right)\cdot 113^{5} + \left(94 a^{2} + 12 a + 109\right)\cdot 113^{6} + \left(82 a^{2} + 10 a + 68\right)\cdot 113^{7} + \left(31 a^{2} + 88 a + 104\right)\cdot 113^{8} + \left(38 a^{2} + 43 a + 5\right)\cdot 113^{9} +O(113^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $12$ |
| $6$ | $2$ | $(2,5)(3,8)$ | $4$ |
| $9$ | $2$ | $(1,6)(2,5)(3,8)(4,7)$ | $-4$ |
| $12$ | $2$ | $(1,4)$ | $-2$ |
| $24$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $36$ | $2$ | $(1,4)(2,3)$ | $0$ |
| $36$ | $2$ | $(1,4)(2,5)(3,8)$ | $-2$ |
| $16$ | $3$ | $(1,6,7)$ | $-3$ |
| $64$ | $3$ | $(1,6,7)(3,5,8)$ | $0$ |
| $12$ | $4$ | $(2,3,5,8)$ | $2$ |
| $36$ | $4$ | $(1,4,6,7)(2,3,5,8)$ | $0$ |
| $36$ | $4$ | $(1,4,6,7)(2,5)(3,8)$ | $2$ |
| $72$ | $4$ | $(1,2,6,5)(3,7,8,4)$ | $0$ |
| $72$ | $4$ | $(1,4)(2,3,5,8)$ | $0$ |
| $144$ | $4$ | $(1,3,4,2)(5,6)(7,8)$ | $0$ |
| $48$ | $6$ | $(1,7,6)(2,5)(3,8)$ | $1$ |
| $96$ | $6$ | $(1,4)(3,8,5)$ | $1$ |
| $192$ | $6$ | $(1,3,6,5,7,8)(2,4)$ | $0$ |
| $144$ | $8$ | $(1,2,4,3,6,5,7,8)$ | $0$ |
| $96$ | $12$ | $(1,6,7)(2,3,5,8)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.