Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 433 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 15 + 285\cdot 433 + 212\cdot 433^{2} + 146\cdot 433^{3} + 413\cdot 433^{4} +O\left(433^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 23 + 184\cdot 433 + 49\cdot 433^{2} + 247\cdot 433^{3} + 80\cdot 433^{4} +O\left(433^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 145 + 43\cdot 433 + 141\cdot 433^{2} + 406\cdot 433^{3} + 226\cdot 433^{4} +O\left(433^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 267 + 419\cdot 433 + 191\cdot 433^{2} + 7\cdot 433^{3} + 420\cdot 433^{4} +O\left(433^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 287 + 377\cdot 433 + 96\cdot 433^{2} + 235\cdot 433^{3} + 209\cdot 433^{4} +O\left(433^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 314 + 418\cdot 433 + 64\cdot 433^{2} + 311\cdot 433^{3} + 413\cdot 433^{4} +O\left(433^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 338 + 98\cdot 433 + 209\cdot 433^{2} + 116\cdot 433^{3} + 257\cdot 433^{4} +O\left(433^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 344 + 337\cdot 433 + 332\cdot 433^{2} + 261\cdot 433^{3} + 143\cdot 433^{4} +O\left(433^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,4,8,5)(2,3,7,6)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
| $(1,7,5,3,8,2,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-2$ |
$-2$ |
| $4$ |
$2$ |
$(1,3)(2,4)(5,7)(6,8)$ |
$0$ |
$0$ |
| $4$ |
$2$ |
$(1,5)(3,6)(4,8)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,4,8,5)(2,3,7,6)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,6,4,2,8,3,5,7)$ |
$-\zeta_{8}^{3} + \zeta_{8}$ |
$\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ |
$8$ |
$(1,2,5,6,8,7,4,3)$ |
$\zeta_{8}^{3} - \zeta_{8}$ |
$-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
