Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 823 }$ to precision 8.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 106 + 663\cdot 823 + 318\cdot 823^{2} + 37\cdot 823^{3} + 138\cdot 823^{4} + 195\cdot 823^{5} + 812\cdot 823^{6} + 723\cdot 823^{7} +O\left(823^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 167 + 751\cdot 823 + 806\cdot 823^{2} + 587\cdot 823^{3} + 601\cdot 823^{4} + 332\cdot 823^{5} + 745\cdot 823^{6} + 594\cdot 823^{7} +O\left(823^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 362 + 234\cdot 823 + 765\cdot 823^{2} + 174\cdot 823^{3} + 314\cdot 823^{4} + 112\cdot 823^{5} + 716\cdot 823^{6} + 173\cdot 823^{7} +O\left(823^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 631 + 162\cdot 823 + 768\cdot 823^{2} + 293\cdot 823^{3} + 7\cdot 823^{4} + 363\cdot 823^{5} + 234\cdot 823^{6} + 793\cdot 823^{7} +O\left(823^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 651 + 308\cdot 823 + 234\cdot 823^{2} + 349\cdot 823^{3} + 657\cdot 823^{4} + 155\cdot 823^{5} + 504\cdot 823^{6} + 498\cdot 823^{7} +O\left(823^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 672 + 742\cdot 823 + 690\cdot 823^{2} + 222\cdot 823^{3} + 477\cdot 823^{4} + 459\cdot 823^{5} + 10\cdot 823^{6} + 421\cdot 823^{7} +O\left(823^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 751 + 655\cdot 823 + 752\cdot 823^{2} + 184\cdot 823^{3} + 211\cdot 823^{4} + 305\cdot 823^{5} + 59\cdot 823^{6} + 163\cdot 823^{7} +O\left(823^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 778 + 595\cdot 823 + 600\cdot 823^{2} + 617\cdot 823^{3} + 61\cdot 823^{4} + 545\cdot 823^{5} + 209\cdot 823^{6} + 746\cdot 823^{7} +O\left(823^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(4,6)(7,8)$ |
| $(1,2)(4,6)$ |
| $(1,8)(2,7)(3,5)(4,6)$ |
| $(1,4,2,6)(3,8,5,7)$ |
| $(3,5)(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $-4$ |
| $2$ | $2$ | $(1,2)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,7)(3,4)(5,6)$ | $0$ |
| $4$ | $2$ | $(1,2)(4,6)$ | $0$ |
| $4$ | $2$ | $(1,8)(2,7)(3,5)(4,6)$ | $2$ |
| $4$ | $2$ | $(1,8)(2,7)$ | $-2$ |
| $4$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $0$ |
| $4$ | $2$ | $(1,4)(2,6)(3,8)(5,7)$ | $0$ |
| $4$ | $4$ | $(1,4,2,6)(3,8,5,7)$ | $0$ |
| $4$ | $4$ | $(1,7,2,8)(3,4,5,6)$ | $0$ |
| $4$ | $4$ | $(1,4,2,6)(3,7,5,8)$ | $0$ |
| $8$ | $4$ | $(1,7,2,8)(3,5)$ | $0$ |
| $8$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |
| $8$ | $4$ | $(1,5,8,6)(2,3,7,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
