Basic invariants
| Dimension: | $56$ |
| Group: | $A_8$ |
| Conductor: | \(253\!\cdots\!136\)\(\medspace = 2^{148} \cdot 3294173^{48} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 8.0.1339918344154038594028110691225010840890507264.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 105 |
| Parity: | even |
| Projective image: | $A_8$ |
| Projective field: | Galois closure of 8.0.1339918344154038594028110691225010840890507264.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 197 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 197 }$:
\( x^{2} + 192x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 59 a + 194 + \left(126 a + 186\right)\cdot 197 + \left(165 a + 86\right)\cdot 197^{2} + \left(51 a + 70\right)\cdot 197^{3} + \left(47 a + 171\right)\cdot 197^{4} + \left(65 a + 39\right)\cdot 197^{5} + \left(146 a + 134\right)\cdot 197^{6} + \left(25 a + 2\right)\cdot 197^{7} + 6 a\cdot 197^{8} + \left(35 a + 55\right)\cdot 197^{9} +O(197^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 138 a + 95 + \left(70 a + 168\right)\cdot 197 + 31 a\cdot 197^{2} + \left(145 a + 164\right)\cdot 197^{3} + \left(149 a + 158\right)\cdot 197^{4} + \left(131 a + 121\right)\cdot 197^{5} + \left(50 a + 12\right)\cdot 197^{6} + \left(171 a + 182\right)\cdot 197^{7} + \left(190 a + 4\right)\cdot 197^{8} + \left(161 a + 27\right)\cdot 197^{9} +O(197^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 111 a + 52 + \left(191 a + 109\right)\cdot 197 + \left(43 a + 4\right)\cdot 197^{2} + \left(155 a + 163\right)\cdot 197^{3} + \left(64 a + 78\right)\cdot 197^{4} + \left(117 a + 32\right)\cdot 197^{5} + \left(169 a + 146\right)\cdot 197^{6} + \left(96 a + 69\right)\cdot 197^{7} + \left(3 a + 148\right)\cdot 197^{8} + 52 a\cdot 197^{9} +O(197^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 86 a + 16 + \left(5 a + 168\right)\cdot 197 + \left(153 a + 32\right)\cdot 197^{2} + \left(41 a + 107\right)\cdot 197^{3} + \left(132 a + 50\right)\cdot 197^{4} + \left(79 a + 160\right)\cdot 197^{5} + \left(27 a + 88\right)\cdot 197^{6} + \left(100 a + 187\right)\cdot 197^{7} + \left(193 a + 68\right)\cdot 197^{8} + \left(144 a + 60\right)\cdot 197^{9} +O(197^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 76 a + 83 + \left(155 a + 143\right)\cdot 197 + \left(196 a + 151\right)\cdot 197^{2} + \left(184 a + 144\right)\cdot 197^{3} + \left(72 a + 167\right)\cdot 197^{4} + \left(70 a + 43\right)\cdot 197^{5} + \left(103 a + 157\right)\cdot 197^{6} + \left(42 a + 150\right)\cdot 197^{7} + \left(178 a + 184\right)\cdot 197^{8} + \left(17 a + 62\right)\cdot 197^{9} +O(197^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 32 a + 158 + \left(53 a + 57\right)\cdot 197 + \left(77 a + 91\right)\cdot 197^{2} + \left(33 a + 80\right)\cdot 197^{3} + \left(74 a + 33\right)\cdot 197^{4} + \left(155 a + 76\right)\cdot 197^{5} + \left(183 a + 130\right)\cdot 197^{6} + \left(158 a + 154\right)\cdot 197^{7} + \left(7 a + 127\right)\cdot 197^{8} + \left(194 a + 19\right)\cdot 197^{9} +O(197^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 165 a + 121 + \left(143 a + 94\right)\cdot 197 + \left(119 a + 30\right)\cdot 197^{2} + \left(163 a + 170\right)\cdot 197^{3} + \left(122 a + 173\right)\cdot 197^{4} + \left(41 a + 187\right)\cdot 197^{5} + \left(13 a + 105\right)\cdot 197^{6} + \left(38 a + 174\right)\cdot 197^{7} + \left(189 a + 7\right)\cdot 197^{8} + \left(2 a + 194\right)\cdot 197^{9} +O(197^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 121 a + 69 + \left(41 a + 56\right)\cdot 197 + 192\cdot 197^{2} + \left(12 a + 84\right)\cdot 197^{3} + \left(124 a + 150\right)\cdot 197^{4} + \left(126 a + 125\right)\cdot 197^{5} + \left(93 a + 12\right)\cdot 197^{6} + \left(154 a + 63\right)\cdot 197^{7} + \left(18 a + 48\right)\cdot 197^{8} + \left(179 a + 171\right)\cdot 197^{9} +O(197^{10})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $56$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $8$ |
| $210$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $112$ | $3$ | $(1,2,3)$ | $-4$ |
| $1120$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $1260$ | $4$ | $(1,2,3,4)(5,6,7,8)$ | $0$ |
| $2520$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $1344$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $1680$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $0$ |
| $3360$ | $6$ | $(1,2,3,4,5,6)(7,8)$ | $-1$ |
| $2880$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $2880$ | $7$ | $(1,3,4,5,6,7,2)$ | $0$ |
| $1344$ | $15$ | $(1,2,3,4,5)(6,7,8)$ | $1$ |
| $1344$ | $15$ | $(1,3,4,5,2)(6,7,8)$ | $1$ |