Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 463 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 115 + 220\cdot 463 + 428\cdot 463^{2} + 74\cdot 463^{3} + 343\cdot 463^{4} +O\left(463^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 202 + 120\cdot 463 + 443\cdot 463^{2} + 142\cdot 463^{3} + 222\cdot 463^{4} +O\left(463^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 368 + 431\cdot 463 + 54\cdot 463^{2} + 285\cdot 463^{3} + 420\cdot 463^{4} +O\left(463^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 391 + 108\cdot 463 + 262\cdot 463^{2} + 371\cdot 463^{3} + 293\cdot 463^{4} +O\left(463^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 404 + 137\cdot 463 + 187\cdot 463^{2} + 313\cdot 463^{3} + 18\cdot 463^{4} +O\left(463^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 416 + 235\cdot 463 + 240\cdot 463^{2} + 184\cdot 463^{3} + 264\cdot 463^{4} +O\left(463^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 432 + 42\cdot 463 + 283\cdot 463^{2} + 2\cdot 463^{3} + 387\cdot 463^{4} +O\left(463^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 452 + 90\cdot 463 + 415\cdot 463^{2} + 13\cdot 463^{3} + 365\cdot 463^{4} +O\left(463^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(2,5)(3,6)$ |
| $(1,2,8,5)(3,7,6,4)$ |
| $(1,8)(2,5)(3,6)(4,7)$ |
| $(1,2,7,3,8,5,4,6)$ |
| $(1,4,8,7)(2,3,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,8)(2,5)(3,6)(4,7)$ |
$-4$ |
| $2$ |
$2$ |
$(2,5)(3,6)$ |
$0$ |
| $4$ |
$2$ |
$(2,3)(4,7)(5,6)$ |
$0$ |
| $4$ |
$2$ |
$(1,4)(2,5)(7,8)$ |
$0$ |
| $4$ |
$2$ |
$(1,3)(2,7)(4,5)(6,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,7,8,4)(2,3,5,6)$ |
$0$ |
| $2$ |
$4$ |
$(1,4,8,7)(2,3,5,6)$ |
$0$ |
| $4$ |
$4$ |
$(1,5,8,2)(3,4,6,7)$ |
$0$ |
| $4$ |
$8$ |
$(1,2,7,3,8,5,4,6)$ |
$0$ |
| $4$ |
$8$ |
$(1,3,7,5,8,6,4,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
