Basic invariants
| Dimension: | $56$ |
| Group: | $A_8$ |
| Conductor: | \(106\!\cdots\!784\)\(\medspace = 2^{144} \cdot 7^{70} \cdot 1075649^{48} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 8.0.2340710530180884465731804242655471841228462751744.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 105 |
| Parity: | even |
| Projective image: | $A_8$ |
| Projective field: | Galois closure of 8.0.2340710530180884465731804242655471841228462751744.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 179 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 179 }$:
\( x^{3} + 4x + 177 \)
![Copy content]()
![Toggle raw display]()
Roots:
| $r_{ 1 }$ | $=$ |
\( 57 + 99\cdot 179 + 26\cdot 179^{2} + 6\cdot 179^{3} + 121\cdot 179^{4} + 61\cdot 179^{5} + 51\cdot 179^{6} + 21\cdot 179^{7} + 126\cdot 179^{8} + 71\cdot 179^{9} +O(179^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 169 + 149\cdot 179 + 6\cdot 179^{2} + 77\cdot 179^{3} + 71\cdot 179^{4} + 154\cdot 179^{5} + 102\cdot 179^{6} + 83\cdot 179^{7} + 144\cdot 179^{8} + 160\cdot 179^{9} +O(179^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 6 a^{2} + 148 a + 40 + \left(19 a^{2} + 66 a + 97\right)\cdot 179 + \left(88 a^{2} + 114 a + 128\right)\cdot 179^{2} + \left(154 a^{2} + 75 a + 16\right)\cdot 179^{3} + \left(131 a^{2} + 167 a + 86\right)\cdot 179^{4} + \left(34 a^{2} + 49 a + 7\right)\cdot 179^{5} + \left(69 a^{2} + 73 a + 120\right)\cdot 179^{6} + \left(101 a^{2} + 123 a + 152\right)\cdot 179^{7} + \left(56 a^{2} + 115 a + 32\right)\cdot 179^{8} + \left(165 a^{2} + 30 a + 55\right)\cdot 179^{9} +O(179^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 25 a^{2} + 8 a + 27 + \left(147 a^{2} + 56 a + 24\right)\cdot 179 + \left(5 a^{2} + 37 a + 51\right)\cdot 179^{2} + \left(47 a^{2} + 135 a + 75\right)\cdot 179^{3} + \left(52 a^{2} + 78 a + 42\right)\cdot 179^{4} + \left(77 a^{2} + 127 a + 40\right)\cdot 179^{5} + \left(163 a^{2} + 133 a + 31\right)\cdot 179^{6} + \left(31 a^{2} + 81 a + 108\right)\cdot 179^{7} + \left(109 a^{2} + 89 a + 20\right)\cdot 179^{8} + \left(34 a^{2} + 16 a + 102\right)\cdot 179^{9} +O(179^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 27 a^{2} + 12 a + 96 + \left(132 a^{2} + 22 a + 100\right)\cdot 179 + \left(62 a^{2} + 56 a + 120\right)\cdot 179^{2} + \left(2 a^{2} + 129 a + 28\right)\cdot 179^{3} + \left(131 a^{2} + 81 a + 24\right)\cdot 179^{4} + \left(131 a^{2} + 56 a + 87\right)\cdot 179^{5} + \left(163 a^{2} + 142 a + 133\right)\cdot 179^{6} + \left(72 a^{2} + 39 a + 76\right)\cdot 179^{7} + \left(121 a^{2} + 123 a + 86\right)\cdot 179^{8} + \left(66 a^{2} + 109 a + 90\right)\cdot 179^{9} +O(179^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 76 a^{2} + 51 a + 163 + \left(166 a^{2} + 177 a + 15\right)\cdot 179 + \left(148 a^{2} + 12 a + 15\right)\cdot 179^{2} + \left(102 a^{2} + 94 a + 45\right)\cdot 179^{3} + \left(142 a^{2} + 40 a + 104\right)\cdot 179^{4} + \left(113 a^{2} + 27 a + 137\right)\cdot 179^{5} + \left(114 a^{2} + 138 a + 139\right)\cdot 179^{6} + \left(68 a^{2} + 44 a + 86\right)\cdot 179^{7} + \left(34 a^{2} + 118 a + 119\right)\cdot 179^{8} + \left(101 a^{2} + 33 a + 100\right)\cdot 179^{9} +O(179^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 78 a^{2} + 120 a + 49 + \left(44 a^{2} + 124 a + 108\right)\cdot 179 + \left(24 a^{2} + 128 a + 40\right)\cdot 179^{2} + \left(29 a^{2} + 128 a + 87\right)\cdot 179^{3} + \left(163 a^{2} + 59 a + 99\right)\cdot 179^{4} + \left(166 a^{2} + 24 a + 40\right)\cdot 179^{5} + \left(79 a^{2} + 86 a + 47\right)\cdot 179^{6} + \left(78 a^{2} + 52 a + 53\right)\cdot 179^{7} + \left(35 a^{2} + 150 a + 122\right)\cdot 179^{8} + \left(43 a^{2} + 128 a + 5\right)\cdot 179^{9} +O(179^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 146 a^{2} + 19 a + 115 + \left(27 a^{2} + 90 a + 120\right)\cdot 179 + \left(28 a^{2} + 8 a + 147\right)\cdot 179^{2} + \left(22 a^{2} + 153 a + 21\right)\cdot 179^{3} + \left(95 a^{2} + 108 a + 167\right)\cdot 179^{4} + \left(12 a^{2} + 72 a + 7\right)\cdot 179^{5} + \left(125 a^{2} + 142 a + 90\right)\cdot 179^{6} + \left(4 a^{2} + 15 a + 133\right)\cdot 179^{7} + \left(a^{2} + 119 a + 63\right)\cdot 179^{8} + \left(126 a^{2} + 38 a + 129\right)\cdot 179^{9} +O(179^{10})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $56$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $8$ |
| $210$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $112$ | $3$ | $(1,2,3)$ | $-4$ |
| $1120$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $1260$ | $4$ | $(1,2,3,4)(5,6,7,8)$ | $0$ |
| $2520$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $1344$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $1680$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $0$ |
| $3360$ | $6$ | $(1,2,3,4,5,6)(7,8)$ | $-1$ |
| $2880$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $2880$ | $7$ | $(1,3,4,5,6,7,2)$ | $0$ |
| $1344$ | $15$ | $(1,2,3,4,5)(6,7,8)$ | $1$ |
| $1344$ | $15$ | $(1,3,4,5,2)(6,7,8)$ | $1$ |