Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 1061 }$ to precision 18.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 46 + 773\cdot 1061 + 180\cdot 1061^{2} + 1000\cdot 1061^{3} + 721\cdot 1061^{4} + 784\cdot 1061^{5} + 388\cdot 1061^{6} + 885\cdot 1061^{7} + 71\cdot 1061^{8} + 1051\cdot 1061^{9} + 544\cdot 1061^{10} + 845\cdot 1061^{11} + 741\cdot 1061^{12} + 90\cdot 1061^{13} + 807\cdot 1061^{14} + 1051\cdot 1061^{15} + 408\cdot 1061^{16} + 269\cdot 1061^{17} +O\left(1061^{ 18 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 120 + 421\cdot 1061 + 695\cdot 1061^{2} + 39\cdot 1061^{3} + 15\cdot 1061^{4} + 652\cdot 1061^{5} + 801\cdot 1061^{6} + 241\cdot 1061^{7} + 271\cdot 1061^{8} + 164\cdot 1061^{9} + 401\cdot 1061^{10} + 367\cdot 1061^{11} + 903\cdot 1061^{12} + 978\cdot 1061^{13} + 10\cdot 1061^{14} + 510\cdot 1061^{15} + 198\cdot 1061^{16} + 954\cdot 1061^{17} +O\left(1061^{ 18 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 280 + 387\cdot 1061 + 117\cdot 1061^{2} + 1032\cdot 1061^{3} + 908\cdot 1061^{4} + 136\cdot 1061^{5} + 416\cdot 1061^{6} + 99\cdot 1061^{7} + 36\cdot 1061^{8} + 861\cdot 1061^{9} + 94\cdot 1061^{10} + 350\cdot 1061^{11} + 993\cdot 1061^{12} + 896\cdot 1061^{13} + 754\cdot 1061^{14} + 698\cdot 1061^{15} + 728\cdot 1061^{16} + 678\cdot 1061^{17} +O\left(1061^{ 18 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 375 + 214\cdot 1061 + 217\cdot 1061^{2} + 833\cdot 1061^{3} + 35\cdot 1061^{4} + 254\cdot 1061^{5} + 1025\cdot 1061^{6} + 576\cdot 1061^{7} + 174\cdot 1061^{8} + 895\cdot 1061^{9} + 607\cdot 1061^{10} + 940\cdot 1061^{11} + 811\cdot 1061^{12} + 23\cdot 1061^{13} + 789\cdot 1061^{14} + 503\cdot 1061^{15} + 546\cdot 1061^{16} + 628\cdot 1061^{17} +O\left(1061^{ 18 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 686 + 846\cdot 1061 + 843\cdot 1061^{2} + 227\cdot 1061^{3} + 1025\cdot 1061^{4} + 806\cdot 1061^{5} + 35\cdot 1061^{6} + 484\cdot 1061^{7} + 886\cdot 1061^{8} + 165\cdot 1061^{9} + 453\cdot 1061^{10} + 120\cdot 1061^{11} + 249\cdot 1061^{12} + 1037\cdot 1061^{13} + 271\cdot 1061^{14} + 557\cdot 1061^{15} + 514\cdot 1061^{16} + 432\cdot 1061^{17} +O\left(1061^{ 18 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 781 + 673\cdot 1061 + 943\cdot 1061^{2} + 28\cdot 1061^{3} + 152\cdot 1061^{4} + 924\cdot 1061^{5} + 644\cdot 1061^{6} + 961\cdot 1061^{7} + 1024\cdot 1061^{8} + 199\cdot 1061^{9} + 966\cdot 1061^{10} + 710\cdot 1061^{11} + 67\cdot 1061^{12} + 164\cdot 1061^{13} + 306\cdot 1061^{14} + 362\cdot 1061^{15} + 332\cdot 1061^{16} + 382\cdot 1061^{17} +O\left(1061^{ 18 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 941 + 639\cdot 1061 + 365\cdot 1061^{2} + 1021\cdot 1061^{3} + 1045\cdot 1061^{4} + 408\cdot 1061^{5} + 259\cdot 1061^{6} + 819\cdot 1061^{7} + 789\cdot 1061^{8} + 896\cdot 1061^{9} + 659\cdot 1061^{10} + 693\cdot 1061^{11} + 157\cdot 1061^{12} + 82\cdot 1061^{13} + 1050\cdot 1061^{14} + 550\cdot 1061^{15} + 862\cdot 1061^{16} + 106\cdot 1061^{17} +O\left(1061^{ 18 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 1015 + 287\cdot 1061 + 880\cdot 1061^{2} + 60\cdot 1061^{3} + 339\cdot 1061^{4} + 276\cdot 1061^{5} + 672\cdot 1061^{6} + 175\cdot 1061^{7} + 989\cdot 1061^{8} + 9\cdot 1061^{9} + 516\cdot 1061^{10} + 215\cdot 1061^{11} + 319\cdot 1061^{12} + 970\cdot 1061^{13} + 253\cdot 1061^{14} + 9\cdot 1061^{15} + 652\cdot 1061^{16} + 791\cdot 1061^{17} +O\left(1061^{ 18 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(2,3)(5,8)(6,7)$ |
| $(4,5)$ |
| $(2,7)$ |
| $(3,6)$ |
| $(1,2)(7,8)$ |
| $(1,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-4$ |
| $2$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $4$ | $2$ | $(3,6)$ | $-2$ |
| $4$ | $2$ | $(2,7)(3,6)$ | $0$ |
| $4$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $4$ | $2$ | $(3,4)(5,6)$ | $2$ |
| $4$ | $2$ | $(1,2)(3,6)(4,5)(7,8)$ | $-2$ |
| $4$ | $2$ | $(1,8)(3,6)(4,5)$ | $2$ |
| $8$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ |
| $8$ | $2$ | $(1,2)(3,6)(7,8)$ | $0$ |
| $4$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ |
| $4$ | $4$ | $(3,5,6,4)$ | $-2$ |
| $4$ | $4$ | $(1,7,8,2)(3,6)(4,5)$ | $2$ |
| $8$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ |
| $8$ | $4$ | $(1,2)(3,5,6,4)(7,8)$ | $0$ |
| $8$ | $4$ | $(1,8)(3,5,6,4)$ | $0$ |
| $16$ | $4$ | $(1,4)(2,3,7,6)(5,8)$ | $0$ |
| $16$ | $4$ | $(1,4,2,3)(5,7,6,8)$ | $0$ |
| $16$ | $8$ | $(1,4,2,3,8,5,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
