Basic invariants
| Dimension: | $9$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(209161465856\)\(\medspace = 2^{12} \cdot 7^{3} \cdot 53^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.14139741952.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 16T1294 |
| Parity: | odd |
| Determinant: | 1.371.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.2.14139741952.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 8x^{5} + 13x^{4} - 18x^{3} - 18x^{2} + 7 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 42 + 73\cdot 113 + 78\cdot 113^{2} + 44\cdot 113^{3} + 47\cdot 113^{4} + 39\cdot 113^{5} + 68\cdot 113^{6} + 105\cdot 113^{7} + 56\cdot 113^{8} + 94\cdot 113^{9} +O(113^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 108 + 26\cdot 113 + 35\cdot 113^{2} + 62\cdot 113^{3} + 74\cdot 113^{4} + 73\cdot 113^{5} + 86\cdot 113^{6} + 68\cdot 113^{7} + 69\cdot 113^{8} + 95\cdot 113^{9} +O(113^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 18 a^{2} + 44 a + 23 + \left(55 a^{2} + 59 a + 59\right)\cdot 113 + \left(2 a^{2} + 20 a + 1\right)\cdot 113^{2} + \left(29 a^{2} + 23 a + 21\right)\cdot 113^{3} + \left(95 a^{2} + 33 a + 31\right)\cdot 113^{4} + \left(34 a^{2} + 2 a + 48\right)\cdot 113^{5} + \left(a^{2} + 91 a + 53\right)\cdot 113^{6} + \left(74 a^{2} + 5 a + 70\right)\cdot 113^{7} + \left(42 a^{2} + 61 a + 53\right)\cdot 113^{8} + \left(17 a^{2} + 91 a + 98\right)\cdot 113^{9} +O(113^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 20 a^{2} + 112 a + 109 + \left(15 a^{2} + 16 a + 71\right)\cdot 113 + \left(17 a^{2} + 79\right)\cdot 113^{2} + \left(12 a^{2} + 36 a + 81\right)\cdot 113^{3} + \left(8 a^{2} + 29 a + 93\right)\cdot 113^{4} + \left(14 a^{2} + 59 a + 12\right)\cdot 113^{5} + \left(70 a^{2} + 5 a + 6\right)\cdot 113^{6} + \left(111 a^{2} + 48 a + 45\right)\cdot 113^{7} + \left(69 a^{2} + 94 a + 86\right)\cdot 113^{8} + \left(86 a^{2} + 22 a + 15\right)\cdot 113^{9} +O(113^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 22 a^{2} + 62 a + 104 + \left(17 a^{2} + 106 a + 104\right)\cdot 113 + \left(81 a^{2} + 72 a + 29\right)\cdot 113^{2} + \left(25 a^{2} + 104 a + 9\right)\cdot 113^{3} + \left(75 a^{2} + 11 a + 84\right)\cdot 113^{4} + \left(92 a^{2} + 10 a + 66\right)\cdot 113^{5} + \left(85 a^{2} + 66 a + 20\right)\cdot 113^{6} + \left(88 a^{2} + 52 a + 99\right)\cdot 113^{7} + \left(78 a^{2} + 14 a + 99\right)\cdot 113^{8} + \left(82 a^{2} + 87 a + 32\right)\cdot 113^{9} +O(113^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 37 a^{2} + 35 a + 71 + \left(9 a^{2} + 5 a + 100\right)\cdot 113 + \left(48 a^{2} + 53 a + 41\right)\cdot 113^{2} + \left(27 a^{2} + 24 a + 18\right)\cdot 113^{3} + \left(75 a^{2} + 66 a + 84\right)\cdot 113^{4} + \left(92 a^{2} + 87 a + 66\right)\cdot 113^{5} + \left(44 a^{2} + 10 a + 65\right)\cdot 113^{6} + \left(43 a^{2} + 99 a + 45\right)\cdot 113^{7} + \left(31 a^{2} + 89 a + 35\right)\cdot 113^{8} + \left(102 a^{2} + 30 a + 24\right)\cdot 113^{9} +O(113^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 54 a^{2} + 16 a + 11 + \left(86 a^{2} + a + 60\right)\cdot 113 + \left(96 a^{2} + 100 a + 75\right)\cdot 113^{2} + \left(59 a^{2} + 96 a + 40\right)\cdot 113^{3} + \left(75 a^{2} + 34 a + 10\right)\cdot 113^{4} + \left(40 a^{2} + 15 a + 53\right)\cdot 113^{5} + \left(95 a^{2} + 36 a + 71\right)\cdot 113^{6} + \left(93 a^{2} + 74 a + 88\right)\cdot 113^{7} + \left(2 a^{2} + 8 a + 33\right)\cdot 113^{8} + \left(41 a^{2} + 108 a + 74\right)\cdot 113^{9} +O(113^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 75 a^{2} + 70 a + 101 + \left(42 a^{2} + 36 a + 67\right)\cdot 113 + \left(93 a^{2} + 92 a + 109\right)\cdot 113^{2} + \left(71 a^{2} + 53 a + 60\right)\cdot 113^{3} + \left(9 a^{2} + 50 a + 26\right)\cdot 113^{4} + \left(64 a^{2} + 51 a + 91\right)\cdot 113^{5} + \left(41 a^{2} + 16 a + 79\right)\cdot 113^{6} + \left(40 a^{2} + 59 a + 41\right)\cdot 113^{7} + \left(70 a + 16\right)\cdot 113^{8} + \left(9 a^{2} + 111 a + 16\right)\cdot 113^{9} +O(113^{10})\)
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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 |
| $1$ | $1$ | $()$ | $9$ |
| $6$ | $2$ | $(2,4)(3,8)$ | $-3$ |
| $9$ | $2$ | $(1,6)(2,4)(3,8)(5,7)$ | $1$ |
| $12$ | $2$ | $(2,3)$ | $3$ |
| $24$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $3$ |
| $36$ | $2$ | $(1,5)(2,3)$ | $1$ |
| $36$ | $2$ | $(1,6)(2,3)(5,7)$ | $-1$ |
| $16$ | $3$ | $(5,7,6)$ | $0$ |
| $64$ | $3$ | $(2,4,8)(5,6,7)$ | $0$ |
| $12$ | $4$ | $(2,3,4,8)$ | $-3$ |
| $36$ | $4$ | $(1,5,6,7)(2,3,4,8)$ | $1$ |
| $36$ | $4$ | $(1,5,6,7)(2,4)(3,8)$ | $1$ |
| $72$ | $4$ | $(1,2,6,4)(3,7,8,5)$ | $-1$ |
| $72$ | $4$ | $(1,5,6,7)(2,3)$ | $-1$ |
| $144$ | $4$ | $(1,2,5,3)(4,6)(7,8)$ | $1$ |
| $48$ | $6$ | $(2,4)(3,8)(5,6,7)$ | $0$ |
| $96$ | $6$ | $(2,3)(5,7,6)$ | $0$ |
| $192$ | $6$ | $(1,3)(2,5,4,6,8,7)$ | $0$ |
| $144$ | $8$ | $(1,2,5,3,6,4,7,8)$ | $-1$ |
| $96$ | $12$ | $(2,3,4,8)(5,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.