Basic invariants
| Dimension: | $56$ |
| Group: | $A_8$ |
| Conductor: | \(514\!\cdots\!576\)\(\medspace = 2^{156} \cdot 23^{48} \cdot 572903^{48} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 8.0.87815967535129608031908549089667529769029306679296.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 105 |
| Parity: | even |
| Projective image: | $A_8$ |
| Projective field: | Galois closure of 8.0.87815967535129608031908549089667529769029306679296.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 263 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 263 }$:
\( x^{2} + 261x + 5 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 210 + 104\cdot 263 + 86\cdot 263^{2} + 48\cdot 263^{3} + 214\cdot 263^{4} + 111\cdot 263^{5} + 131\cdot 263^{6} + 248\cdot 263^{7} + 193\cdot 263^{8} + 16\cdot 263^{9} +O(263^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 75 a + 168 + \left(192 a + 132\right)\cdot 263 + \left(217 a + 99\right)\cdot 263^{2} + \left(35 a + 65\right)\cdot 263^{3} + \left(141 a + 160\right)\cdot 263^{4} + \left(211 a + 25\right)\cdot 263^{5} + \left(126 a + 103\right)\cdot 263^{6} + \left(170 a + 99\right)\cdot 263^{7} + \left(142 a + 158\right)\cdot 263^{8} + \left(249 a + 136\right)\cdot 263^{9} +O(263^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 45 + 43\cdot 263 + 214\cdot 263^{2} + 253\cdot 263^{3} + 80\cdot 263^{4} + 21\cdot 263^{5} + 169\cdot 263^{6} + 147\cdot 263^{7} + 211\cdot 263^{8} + 189\cdot 263^{9} +O(263^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 179 a + 69 + \left(43 a + 73\right)\cdot 263 + \left(260 a + 36\right)\cdot 263^{2} + \left(232 a + 184\right)\cdot 263^{3} + \left(131 a + 158\right)\cdot 263^{4} + \left(166 a + 99\right)\cdot 263^{5} + \left(134 a + 57\right)\cdot 263^{6} + \left(255 a + 236\right)\cdot 263^{7} + \left(107 a + 146\right)\cdot 263^{8} + \left(68 a + 17\right)\cdot 263^{9} +O(263^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 84 a + 164 + \left(219 a + 244\right)\cdot 263 + \left(2 a + 249\right)\cdot 263^{2} + \left(30 a + 126\right)\cdot 263^{3} + \left(131 a + 189\right)\cdot 263^{4} + \left(96 a + 37\right)\cdot 263^{5} + \left(128 a + 160\right)\cdot 263^{6} + \left(7 a + 86\right)\cdot 263^{7} + \left(155 a + 107\right)\cdot 263^{8} + \left(194 a + 46\right)\cdot 263^{9} +O(263^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 184 + 217\cdot 263 + 6\cdot 263^{2} + 92\cdot 263^{3} + 48\cdot 263^{4} + 92\cdot 263^{5} + 262\cdot 263^{6} + 65\cdot 263^{7} + 47\cdot 263^{8} + 2\cdot 263^{9} +O(263^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 188 a + 55 + \left(70 a + 179\right)\cdot 263 + \left(45 a + 79\right)\cdot 263^{2} + \left(227 a + 182\right)\cdot 263^{3} + \left(121 a + 143\right)\cdot 263^{4} + \left(51 a + 44\right)\cdot 263^{5} + \left(136 a + 145\right)\cdot 263^{6} + \left(92 a + 50\right)\cdot 263^{7} + \left(120 a + 10\right)\cdot 263^{8} + \left(13 a + 230\right)\cdot 263^{9} +O(263^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 157 + 56\cdot 263 + 16\cdot 263^{2} + 99\cdot 263^{3} + 56\cdot 263^{4} + 93\cdot 263^{5} + 23\cdot 263^{6} + 117\cdot 263^{7} + 176\cdot 263^{8} + 149\cdot 263^{9} +O(263^{10})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $56$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $8$ |
| $210$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $112$ | $3$ | $(1,2,3)$ | $-4$ |
| $1120$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $1260$ | $4$ | $(1,2,3,4)(5,6,7,8)$ | $0$ |
| $2520$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $1344$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $1680$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $0$ |
| $3360$ | $6$ | $(1,2,3,4,5,6)(7,8)$ | $-1$ |
| $2880$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $2880$ | $7$ | $(1,3,4,5,6,7,2)$ | $0$ |
| $1344$ | $15$ | $(1,2,3,4,5)(6,7,8)$ | $1$ |
| $1344$ | $15$ | $(1,3,4,5,2)(6,7,8)$ | $1$ |