Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 461 }$ to precision 8.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 72 + 210\cdot 461 + 205\cdot 461^{2} + 101\cdot 461^{3} + 327\cdot 461^{4} + 64\cdot 461^{5} + 271\cdot 461^{6} + 268\cdot 461^{7} +O\left(461^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 125 + 345\cdot 461 + 152\cdot 461^{2} + 183\cdot 461^{3} + 209\cdot 461^{4} + 135\cdot 461^{5} + 319\cdot 461^{6} + 108\cdot 461^{7} +O\left(461^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 150 + 72\cdot 461 + 218\cdot 461^{2} + 173\cdot 461^{3} + 186\cdot 461^{4} + 146\cdot 461^{5} + 212\cdot 461^{6} + 5\cdot 461^{7} +O\left(461^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 181 + 251\cdot 461 + 201\cdot 461^{2} + 222\cdot 461^{3} + 357\cdot 461^{4} + 326\cdot 461^{5} + 240\cdot 461^{6} + 413\cdot 461^{7} +O\left(461^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 280 + 209\cdot 461 + 259\cdot 461^{2} + 238\cdot 461^{3} + 103\cdot 461^{4} + 134\cdot 461^{5} + 220\cdot 461^{6} + 47\cdot 461^{7} +O\left(461^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 311 + 388\cdot 461 + 242\cdot 461^{2} + 287\cdot 461^{3} + 274\cdot 461^{4} + 314\cdot 461^{5} + 248\cdot 461^{6} + 455\cdot 461^{7} +O\left(461^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 336 + 115\cdot 461 + 308\cdot 461^{2} + 277\cdot 461^{3} + 251\cdot 461^{4} + 325\cdot 461^{5} + 141\cdot 461^{6} + 352\cdot 461^{7} +O\left(461^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 389 + 250\cdot 461 + 255\cdot 461^{2} + 359\cdot 461^{3} + 133\cdot 461^{4} + 396\cdot 461^{5} + 189\cdot 461^{6} + 192\cdot 461^{7} +O\left(461^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(2,7)(4,5)$ |
| $(1,4,8,5)(2,3,7,6)$ |
| $(2,4)(3,6)(5,7)$ |
| $(1,6,8,3)(2,5,7,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-4$ |
| $2$ |
$2$ |
$(2,7)(4,5)$ |
$0$ |
| $4$ |
$2$ |
$(1,7)(2,8)(3,5)(4,6)$ |
$0$ |
| $4$ |
$2$ |
$(2,4)(3,6)(5,7)$ |
$0$ |
| $4$ |
$2$ |
$(1,6)(3,8)(4,5)$ |
$0$ |
| $2$ |
$4$ |
$(1,6,8,3)(2,5,7,4)$ |
$0$ |
| $2$ |
$4$ |
$(1,6,8,3)(2,4,7,5)$ |
$0$ |
| $4$ |
$4$ |
$(1,4,8,5)(2,3,7,6)$ |
$0$ |
| $4$ |
$8$ |
$(1,2,6,4,8,7,3,5)$ |
$0$ |
| $4$ |
$8$ |
$(1,7,3,4,8,2,6,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
