Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 457 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 15 + 123\cdot 457 + 149\cdot 457^{2} + 50\cdot 457^{3} + 407\cdot 457^{4} +O\left(457^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 23 + 243\cdot 457 + 41\cdot 457^{2} + 433\cdot 457^{3} + 430\cdot 457^{4} +O\left(457^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 130 + 360\cdot 457 + 146\cdot 457^{2} + 138\cdot 457^{3} + 454\cdot 457^{4} +O\left(457^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 230 + 26\cdot 457 + 4\cdot 457^{2} + 347\cdot 457^{3} + 277\cdot 457^{4} +O\left(457^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 301 + 39\cdot 457 + 129\cdot 457^{2} + 67\cdot 457^{3} + 336\cdot 457^{4} +O\left(457^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 306 + 251\cdot 457 + 40\cdot 457^{2} + 381\cdot 457^{3} + 274\cdot 457^{4} +O\left(457^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 383 + 219\cdot 457 + 139\cdot 457^{2} + 134\cdot 457^{3} + 436\cdot 457^{4} +O\left(457^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 442 + 106\cdot 457 + 263\cdot 457^{2} + 276\cdot 457^{3} + 124\cdot 457^{4} +O\left(457^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(2,6)(3,5)(7,8)$ |
| $(1,8)(2,4)(3,6)(5,7)$ |
| $(1,3)(2,7)(4,5)(6,8)$ |
| $(1,2,4,6)(3,7,5,8)$ |
| $(1,3)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,3)(2,7)(4,5)(6,8)$ |
$-4$ |
| $2$ |
$2$ |
$(1,4)(2,6)(3,5)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,3)(4,5)$ |
$0$ |
| $2$ |
$2$ |
$(1,5)(2,6)(3,4)(7,8)$ |
$0$ |
| $4$ |
$2$ |
$(1,8)(2,4)(3,6)(5,7)$ |
$0$ |
| $4$ |
$4$ |
$(1,2,4,6)(3,7,5,8)$ |
$0$ |
| $4$ |
$4$ |
$(1,6,4,2)(3,8,5,7)$ |
$0$ |
| $4$ |
$4$ |
$(1,2,3,7)(4,8,5,6)$ |
$0$ |
| $4$ |
$4$ |
$(1,5,3,4)(6,8)$ |
$0$ |
| $4$ |
$4$ |
$(1,4,3,5)(6,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
