Basic invariants
| Dimension: | $70$ |
| Group: | $A_8$ |
| Conductor: | \(343\!\cdots\!456\)\(\medspace = 2^{188} \cdot 11^{60} \cdot 74869^{60} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 8.0.1277992348243533546275162547509832257536.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 120 |
| Parity: | even |
| Projective image: | $A_8$ |
| Projective field: | Galois closure of 8.0.1277992348243533546275162547509832257536.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 787 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 787 }$:
\( x^{2} + 786x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 215 a + 118 + \left(2 a + 494\right)\cdot 787 + \left(682 a + 42\right)\cdot 787^{2} + \left(634 a + 245\right)\cdot 787^{3} + \left(135 a + 160\right)\cdot 787^{4} + \left(232 a + 190\right)\cdot 787^{5} + \left(721 a + 681\right)\cdot 787^{6} + \left(148 a + 636\right)\cdot 787^{7} + \left(204 a + 757\right)\cdot 787^{8} + \left(112 a + 491\right)\cdot 787^{9} +O(787^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 486 a + 233 + \left(280 a + 599\right)\cdot 787 + \left(656 a + 205\right)\cdot 787^{2} + \left(143 a + 44\right)\cdot 787^{3} + \left(552 a + 302\right)\cdot 787^{4} + \left(588 a + 569\right)\cdot 787^{5} + \left(746 a + 150\right)\cdot 787^{6} + \left(304 a + 338\right)\cdot 787^{7} + \left(359 a + 124\right)\cdot 787^{8} + \left(249 a + 666\right)\cdot 787^{9} +O(787^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 637 a + 523 + \left(10 a + 20\right)\cdot 787 + \left(542 a + 560\right)\cdot 787^{2} + \left(134 a + 523\right)\cdot 787^{3} + \left(115 a + 52\right)\cdot 787^{4} + \left(440 a + 715\right)\cdot 787^{5} + \left(739 a + 9\right)\cdot 787^{6} + \left(33 a + 448\right)\cdot 787^{7} + \left(229 a + 709\right)\cdot 787^{8} + \left(165 a + 592\right)\cdot 787^{9} +O(787^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 572 a + 333 + \left(784 a + 281\right)\cdot 787 + \left(104 a + 722\right)\cdot 787^{2} + \left(152 a + 197\right)\cdot 787^{3} + \left(651 a + 448\right)\cdot 787^{4} + \left(554 a + 286\right)\cdot 787^{5} + \left(65 a + 383\right)\cdot 787^{6} + \left(638 a + 64\right)\cdot 787^{7} + \left(582 a + 26\right)\cdot 787^{8} + \left(674 a + 400\right)\cdot 787^{9} +O(787^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 394 a + 621 + \left(764 a + 9\right)\cdot 787 + \left(784 a + 749\right)\cdot 787^{2} + \left(780 a + 65\right)\cdot 787^{3} + \left(220 a + 607\right)\cdot 787^{4} + \left(254 a + 640\right)\cdot 787^{5} + \left(748 a + 11\right)\cdot 787^{6} + \left(315 a + 440\right)\cdot 787^{7} + \left(456 a + 546\right)\cdot 787^{8} + \left(152 a + 107\right)\cdot 787^{9} +O(787^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 393 a + 228 + \left(22 a + 380\right)\cdot 787 + \left(2 a + 769\right)\cdot 787^{2} + \left(6 a + 61\right)\cdot 787^{3} + \left(566 a + 47\right)\cdot 787^{4} + \left(532 a + 674\right)\cdot 787^{5} + \left(38 a + 505\right)\cdot 787^{6} + \left(471 a + 7\right)\cdot 787^{7} + \left(330 a + 687\right)\cdot 787^{8} + \left(634 a + 590\right)\cdot 787^{9} +O(787^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 150 a + 373 + \left(776 a + 181\right)\cdot 787 + \left(244 a + 304\right)\cdot 787^{2} + \left(652 a + 116\right)\cdot 787^{3} + \left(671 a + 33\right)\cdot 787^{4} + \left(346 a + 253\right)\cdot 787^{5} + \left(47 a + 309\right)\cdot 787^{6} + \left(753 a + 529\right)\cdot 787^{7} + \left(557 a + 117\right)\cdot 787^{8} + \left(621 a + 529\right)\cdot 787^{9} +O(787^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 301 a + 719 + \left(506 a + 393\right)\cdot 787 + \left(130 a + 581\right)\cdot 787^{2} + \left(643 a + 318\right)\cdot 787^{3} + \left(234 a + 710\right)\cdot 787^{4} + \left(198 a + 605\right)\cdot 787^{5} + \left(40 a + 308\right)\cdot 787^{6} + \left(482 a + 683\right)\cdot 787^{7} + \left(427 a + 178\right)\cdot 787^{8} + \left(537 a + 556\right)\cdot 787^{9} +O(787^{10})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $70$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $-2$ |
| $210$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $112$ | $3$ | $(1,2,3)$ | $-5$ |
| $1120$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
| $1260$ | $4$ | $(1,2,3,4)(5,6,7,8)$ | $-2$ |
| $2520$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $1344$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $1680$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $3360$ | $6$ | $(1,2,3,4,5,6)(7,8)$ | $1$ |
| $2880$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $2880$ | $7$ | $(1,3,4,5,6,7,2)$ | $0$ |
| $1344$ | $15$ | $(1,2,3,4,5)(6,7,8)$ | $0$ |
| $1344$ | $15$ | $(1,3,4,5,2)(6,7,8)$ | $0$ |