Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 463 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 46 + 76\cdot 463 + 119\cdot 463^{2} + 143\cdot 463^{3} + 339\cdot 463^{4} + 328\cdot 463^{5} + 138\cdot 463^{6} +O\left(463^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 95 + 29\cdot 463 + 263\cdot 463^{2} + 365\cdot 463^{3} + 28\cdot 463^{4} + 356\cdot 463^{5} + 113\cdot 463^{6} +O\left(463^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 122 + 441\cdot 463 + 377\cdot 463^{2} + 389\cdot 463^{3} + 383\cdot 463^{4} + 43\cdot 463^{5} + 397\cdot 463^{6} +O\left(463^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 206 + 47\cdot 463 + 78\cdot 463^{2} + 65\cdot 463^{3} + 348\cdot 463^{4} + 336\cdot 463^{5} + 366\cdot 463^{6} +O\left(463^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 244 + 118\cdot 463 + 374\cdot 463^{2} + 162\cdot 463^{3} + 133\cdot 463^{4} + 450\cdot 463^{5} + 6\cdot 463^{6} +O\left(463^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 338 + 139\cdot 463 + 385\cdot 463^{2} + 254\cdot 463^{3} + 27\cdot 463^{4} + 383\cdot 463^{5} + 15\cdot 463^{6} +O\left(463^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 391 + 213\cdot 463 + 16\cdot 463^{2} + 22\cdot 463^{3} + 120\cdot 463^{4} + 313\cdot 463^{5} + 353\cdot 463^{6} +O\left(463^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 412 + 322\cdot 463 + 237\cdot 463^{2} + 448\cdot 463^{3} + 7\cdot 463^{4} + 103\cdot 463^{5} + 459\cdot 463^{6} +O\left(463^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,8)(4,7)(5,6)$ |
| $(1,2)(6,7)$ |
| $(1,2)(4,5)$ |
| $(1,5,2,4)(3,6,8,7)$ |
| $(1,7,2,6)$ |
| $(3,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,2)(3,8)(4,5)(6,7)$ |
$-4$ |
| $2$ |
$2$ |
$(1,2)(6,7)$ |
$0$ |
| $4$ |
$2$ |
$(1,3)(2,8)(4,7)(5,6)$ |
$0$ |
| $4$ |
$2$ |
$(3,8)(6,7)$ |
$0$ |
| $4$ |
$2$ |
$(1,8)(2,3)(4,7)(5,6)$ |
$0$ |
| $4$ |
$2$ |
$(1,7)(2,6)(3,5)(4,8)$ |
$0$ |
| $8$ |
$2$ |
$(1,6)(2,7)(3,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,6,2,7)(3,5,8,4)$ |
$0$ |
| $2$ |
$4$ |
$(1,7,2,6)(3,5,8,4)$ |
$0$ |
| $4$ |
$4$ |
$(1,5,2,4)(3,6,8,7)$ |
$0$ |
| $4$ |
$4$ |
$(1,8,2,3)(4,6,5,7)$ |
$0$ |
| $4$ |
$4$ |
$(1,7,2,6)$ |
$2$ |
| $4$ |
$4$ |
$(1,2)(3,4,8,5)(6,7)$ |
$-2$ |
| $8$ |
$8$ |
$(1,4,7,8,2,5,6,3)$ |
$0$ |
| $8$ |
$8$ |
$(1,4,6,8,2,5,7,3)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
