Basic invariants
| Dimension: | $56$ |
| Group: | $A_8$ |
| Conductor: | \(514\!\cdots\!656\)\(\medspace = 2^{156} \cdot 11^{48} \cdot 151^{48} \cdot 7933^{48} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 8.0.87813728302462049260764173611685476867240408121344.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 105 |
| Parity: | even |
| Projective image: | $A_8$ |
| Projective field: | Galois closure of 8.0.87813728302462049260764173611685476867240408121344.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 157 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 157 }$:
\( x^{2} + 152x + 5 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 22 + 125\cdot 157 + 132\cdot 157^{2} + 119\cdot 157^{3} + 141\cdot 157^{4} + 34\cdot 157^{5} + 19\cdot 157^{6} + 121\cdot 157^{7} + 141\cdot 157^{8} + 129\cdot 157^{9} +O(157^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 106 a + 115 + \left(77 a + 101\right)\cdot 157 + \left(13 a + 85\right)\cdot 157^{2} + \left(38 a + 155\right)\cdot 157^{3} + \left(127 a + 126\right)\cdot 157^{4} + \left(29 a + 88\right)\cdot 157^{5} + \left(52 a + 121\right)\cdot 157^{6} + \left(104 a + 114\right)\cdot 157^{7} + \left(59 a + 79\right)\cdot 157^{8} + \left(40 a + 67\right)\cdot 157^{9} +O(157^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 103 a + 100 + \left(155 a + 22\right)\cdot 157 + \left(87 a + 51\right)\cdot 157^{2} + \left(112 a + 106\right)\cdot 157^{3} + \left(74 a + 121\right)\cdot 157^{4} + \left(27 a + 93\right)\cdot 157^{5} + \left(98 a + 80\right)\cdot 157^{6} + \left(60 a + 51\right)\cdot 157^{7} + \left(125 a + 7\right)\cdot 157^{8} + \left(6 a + 144\right)\cdot 157^{9} +O(157^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 124 + 154\cdot 157 + 97\cdot 157^{2} + 27\cdot 157^{3} + 5\cdot 157^{4} + 157^{5} + 22\cdot 157^{6} + 64\cdot 157^{7} + 128\cdot 157^{8} + 148\cdot 157^{9} +O(157^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 117 + 111\cdot 157 + 145\cdot 157^{2} + 32\cdot 157^{3} + 49\cdot 157^{4} + 27\cdot 157^{5} + 121\cdot 157^{6} + 32\cdot 157^{7} + 34\cdot 157^{8} + 155\cdot 157^{9} +O(157^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 54 a + 144 + \left(a + 69\right)\cdot 157 + \left(69 a + 21\right)\cdot 157^{2} + \left(44 a + 110\right)\cdot 157^{3} + \left(82 a + 68\right)\cdot 157^{4} + \left(129 a + 156\right)\cdot 157^{5} + \left(58 a + 72\right)\cdot 157^{6} + \left(96 a + 99\right)\cdot 157^{7} + \left(31 a + 102\right)\cdot 157^{8} + \left(150 a + 52\right)\cdot 157^{9} +O(157^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 146 + 128\cdot 157 + 17\cdot 157^{2} + 57\cdot 157^{3} + 17\cdot 157^{4} + 115\cdot 157^{5} + 151\cdot 157^{6} + 30\cdot 157^{7} + 17\cdot 157^{8} + 34\cdot 157^{9} +O(157^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 51 a + 17 + \left(79 a + 70\right)\cdot 157 + \left(143 a + 75\right)\cdot 157^{2} + \left(118 a + 18\right)\cdot 157^{3} + \left(29 a + 97\right)\cdot 157^{4} + \left(127 a + 110\right)\cdot 157^{5} + \left(104 a + 38\right)\cdot 157^{6} + \left(52 a + 113\right)\cdot 157^{7} + \left(97 a + 116\right)\cdot 157^{8} + \left(116 a + 52\right)\cdot 157^{9} +O(157^{10})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $56$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $8$ |
| $210$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $112$ | $3$ | $(1,2,3)$ | $-4$ |
| $1120$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $1260$ | $4$ | $(1,2,3,4)(5,6,7,8)$ | $0$ |
| $2520$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $1344$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $1680$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $0$ |
| $3360$ | $6$ | $(1,2,3,4,5,6)(7,8)$ | $-1$ |
| $2880$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $2880$ | $7$ | $(1,3,4,5,6,7,2)$ | $0$ |
| $1344$ | $15$ | $(1,2,3,4,5)(6,7,8)$ | $1$ |
| $1344$ | $15$ | $(1,3,4,5,2)(6,7,8)$ | $1$ |