Basic invariants
| Dimension: | $64$ |
| Group: | $A_8$ |
| Conductor: | \(239\!\cdots\!984\)\(\medspace = 2^{184} \cdot 29^{54} \cdot 3917^{54} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.2252730971538337484304305478905626624.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 168 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $A_8$ |
| Projective stem field: | Galois closure of 8.0.2252730971538337484304305478905626624.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 112x^{6} - 896x^{5} - 3360x^{4} - 7168x^{3} - 8960x^{2} - 6144x + 210826816 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 199 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 199 }$:
\( x^{2} + 193x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 184 a + 142 + \left(180 a + 137\right)\cdot 199 + \left(192 a + 196\right)\cdot 199^{2} + \left(42 a + 173\right)\cdot 199^{3} + \left(21 a + 134\right)\cdot 199^{4} + \left(157 a + 87\right)\cdot 199^{5} + \left(179 a + 76\right)\cdot 199^{6} + \left(164 a + 79\right)\cdot 199^{7} + \left(65 a + 185\right)\cdot 199^{8} + \left(156 a + 130\right)\cdot 199^{9} +O(199^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 23 a + 187 + \left(192 a + 72\right)\cdot 199 + \left(57 a + 161\right)\cdot 199^{2} + \left(5 a + 110\right)\cdot 199^{3} + \left(194 a + 24\right)\cdot 199^{4} + \left(194 a + 89\right)\cdot 199^{5} + \left(5 a + 130\right)\cdot 199^{6} + \left(96 a + 69\right)\cdot 199^{7} + \left(67 a + 92\right)\cdot 199^{8} + \left(71 a + 1\right)\cdot 199^{9} +O(199^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 6 a + 192 + \left(143 a + 28\right)\cdot 199 + \left(154 a + 147\right)\cdot 199^{2} + \left(34 a + 183\right)\cdot 199^{3} + \left(146 a + 109\right)\cdot 199^{4} + \left(185 a + 165\right)\cdot 199^{5} + \left(25 a + 160\right)\cdot 199^{6} + \left(49 a + 48\right)\cdot 199^{7} + \left(128 a + 3\right)\cdot 199^{8} + \left(63 a + 68\right)\cdot 199^{9} +O(199^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 183 a + 82 + \left(164 a + 104\right)\cdot 199 + \left(118 a + 152\right)\cdot 199^{2} + \left(131 a + 144\right)\cdot 199^{3} + \left(19 a + 87\right)\cdot 199^{4} + \left(143 a + 193\right)\cdot 199^{5} + \left(78 a + 195\right)\cdot 199^{6} + \left(88 a + 144\right)\cdot 199^{7} + \left(65 a + 31\right)\cdot 199^{8} + \left(68 a + 78\right)\cdot 199^{9} +O(199^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 193 a + 29 + \left(55 a + 85\right)\cdot 199 + \left(44 a + 136\right)\cdot 199^{2} + \left(164 a + 38\right)\cdot 199^{3} + \left(52 a + 156\right)\cdot 199^{4} + \left(13 a + 138\right)\cdot 199^{5} + \left(173 a + 130\right)\cdot 199^{6} + \left(149 a + 118\right)\cdot 199^{7} + \left(70 a + 126\right)\cdot 199^{8} + \left(135 a + 122\right)\cdot 199^{9} +O(199^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 16 a + 185 + \left(34 a + 114\right)\cdot 199 + \left(80 a + 103\right)\cdot 199^{2} + \left(67 a + 19\right)\cdot 199^{3} + \left(179 a + 74\right)\cdot 199^{4} + \left(55 a + 37\right)\cdot 199^{5} + \left(120 a + 127\right)\cdot 199^{6} + \left(110 a + 198\right)\cdot 199^{7} + \left(133 a + 136\right)\cdot 199^{8} + \left(130 a + 24\right)\cdot 199^{9} +O(199^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 15 a + 52 + \left(18 a + 44\right)\cdot 199 + \left(6 a + 178\right)\cdot 199^{2} + \left(156 a + 39\right)\cdot 199^{3} + \left(177 a + 20\right)\cdot 199^{4} + \left(41 a + 14\right)\cdot 199^{5} + \left(19 a + 3\right)\cdot 199^{6} + \left(34 a + 93\right)\cdot 199^{7} + \left(133 a + 17\right)\cdot 199^{8} + \left(42 a + 8\right)\cdot 199^{9} +O(199^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 176 a + 126 + \left(6 a + 8\right)\cdot 199 + \left(141 a + 118\right)\cdot 199^{2} + \left(193 a + 84\right)\cdot 199^{3} + \left(4 a + 188\right)\cdot 199^{4} + \left(4 a + 69\right)\cdot 199^{5} + \left(193 a + 170\right)\cdot 199^{6} + \left(102 a + 42\right)\cdot 199^{7} + \left(131 a + 3\right)\cdot 199^{8} + \left(127 a + 163\right)\cdot 199^{9} +O(199^{10})\)
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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 |
| $1$ | $1$ | $()$ | $64$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $210$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $112$ | $3$ | $(1,2,3)$ | $4$ |
| $1120$ | $3$ | $(1,2,3)(4,5,6)$ | $-2$ |
| $1260$ | $4$ | $(1,2,3,4)(5,6,7,8)$ | $0$ |
| $2520$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $1344$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $1680$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $0$ |
| $3360$ | $6$ | $(1,2,3,4,5,6)(7,8)$ | $0$ |
| $2880$ | $7$ | $(1,2,3,4,5,6,7)$ | $1$ |
| $2880$ | $7$ | $(1,3,4,5,6,7,2)$ | $1$ |
| $1344$ | $15$ | $(1,2,3,4,5)(6,7,8)$ | $-1$ |
| $1344$ | $15$ | $(1,3,4,5,2)(6,7,8)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.