Basic invariants
| Dimension: | $9$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(801628120832\)\(\medspace = 2^{8} \cdot 7^{3} \cdot 11^{3} \cdot 19^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 8.2.170984391424.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 16T1294 |
| Parity: | odd |
| Projective image: | $S_4\wr C_2$ |
| Projective field: | Galois closure of 8.2.170984391424.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 113 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 113 }$:
\( x^{2} + 101x + 3 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 93 a + 62 + \left(29 a + 61\right)\cdot 113 + \left(92 a + 75\right)\cdot 113^{2} + \left(64 a + 60\right)\cdot 113^{3} + \left(32 a + 38\right)\cdot 113^{4} + \left(34 a + 42\right)\cdot 113^{5} + \left(99 a + 25\right)\cdot 113^{6} + \left(18 a + 63\right)\cdot 113^{7} + \left(13 a + 35\right)\cdot 113^{8} + \left(17 a + 42\right)\cdot 113^{9} +O(113^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 84 + 20\cdot 113 + 107\cdot 113^{2} + 104\cdot 113^{3} + 98\cdot 113^{4} + 30\cdot 113^{5} + 33\cdot 113^{6} + 44\cdot 113^{7} + 88\cdot 113^{8} + 98\cdot 113^{9} +O(113^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 71 a + 75 + \left(6 a + 40\right)\cdot 113 + \left(66 a + 74\right)\cdot 113^{2} + \left(77 a + 37\right)\cdot 113^{3} + \left(36 a + 88\right)\cdot 113^{4} + \left(75 a + 19\right)\cdot 113^{5} + \left(12 a + 87\right)\cdot 113^{6} + \left(38 a + 32\right)\cdot 113^{7} + \left(53 a + 22\right)\cdot 113^{8} + \left(54 a + 2\right)\cdot 113^{9} +O(113^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 42 a + 23 + \left(106 a + 49\right)\cdot 113 + \left(46 a + 69\right)\cdot 113^{2} + \left(35 a + 111\right)\cdot 113^{3} + \left(76 a + 111\right)\cdot 113^{4} + \left(37 a + 95\right)\cdot 113^{5} + \left(100 a + 50\right)\cdot 113^{6} + \left(74 a + 25\right)\cdot 113^{7} + \left(59 a + 59\right)\cdot 113^{8} + \left(58 a + 37\right)\cdot 113^{9} +O(113^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 46 + 2\cdot 113 + 88\cdot 113^{2} + 84\cdot 113^{3} + 39\cdot 113^{4} + 79\cdot 113^{5} + 54\cdot 113^{6} + 10\cdot 113^{7} + 56\cdot 113^{8} + 87\cdot 113^{9} +O(113^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 79 a + 37 + \left(8 a + 19\right)\cdot 113 + \left(55 a + 20\right)\cdot 113^{2} + \left(59 a + 58\right)\cdot 113^{3} + \left(88 a + 31\right)\cdot 113^{4} + \left(76 a + 86\right)\cdot 113^{5} + \left(14 a + 80\right)\cdot 113^{6} + \left(36 a + 2\right)\cdot 113^{7} + \left(22 a + 5\right)\cdot 113^{8} + 101 a\cdot 113^{9} +O(113^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 34 a + 81 + \left(104 a + 44\right)\cdot 113 + \left(57 a + 107\right)\cdot 113^{2} + \left(53 a + 38\right)\cdot 113^{3} + \left(24 a + 17\right)\cdot 113^{4} + \left(36 a + 15\right)\cdot 113^{5} + \left(98 a + 67\right)\cdot 113^{6} + \left(76 a + 82\right)\cdot 113^{7} + \left(90 a + 10\right)\cdot 113^{8} + \left(11 a + 62\right)\cdot 113^{9} +O(113^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 20 a + 48 + \left(83 a + 100\right)\cdot 113 + \left(20 a + 22\right)\cdot 113^{2} + \left(48 a + 68\right)\cdot 113^{3} + \left(80 a + 25\right)\cdot 113^{4} + \left(78 a + 82\right)\cdot 113^{5} + \left(13 a + 52\right)\cdot 113^{6} + \left(94 a + 77\right)\cdot 113^{7} + \left(99 a + 61\right)\cdot 113^{8} + \left(95 a + 8\right)\cdot 113^{9} +O(113^{10})\)
|
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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $9$ |
| $6$ | $2$ | $(2,4)(3,5)$ | $-3$ |
| $9$ | $2$ | $(1,7)(2,4)(3,5)(6,8)$ | $1$ |
| $12$ | $2$ | $(2,3)$ | $3$ |
| $24$ | $2$ | $(1,2)(3,6)(4,7)(5,8)$ | $3$ |
| $36$ | $2$ | $(1,6)(2,3)$ | $1$ |
| $36$ | $2$ | $(1,7)(2,3)(6,8)$ | $-1$ |
| $16$ | $3$ | $(6,8,7)$ | $0$ |
| $64$ | $3$ | $(2,4,5)(6,7,8)$ | $0$ |
| $12$ | $4$ | $(2,3,4,5)$ | $-3$ |
| $36$ | $4$ | $(1,6,7,8)(2,3,4,5)$ | $1$ |
| $36$ | $4$ | $(1,6,7,8)(2,4)(3,5)$ | $1$ |
| $72$ | $4$ | $(1,2,7,4)(3,8,5,6)$ | $-1$ |
| $72$ | $4$ | $(1,6,7,8)(2,3)$ | $-1$ |
| $144$ | $4$ | $(1,2,6,3)(4,7)(5,8)$ | $1$ |
| $48$ | $6$ | $(2,4)(3,5)(6,7,8)$ | $0$ |
| $96$ | $6$ | $(2,3)(6,8,7)$ | $0$ |
| $192$ | $6$ | $(1,3)(2,6,4,7,5,8)$ | $0$ |
| $144$ | $8$ | $(1,2,6,3,7,4,8,5)$ | $-1$ |
| $96$ | $12$ | $(2,3,4,5)(6,8,7)$ | $0$ |