Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 433 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 32 + 9\cdot 433 + 89\cdot 433^{2} + 300\cdot 433^{3} + 336\cdot 433^{4} +O\left(433^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 54 + 84\cdot 433 + 102\cdot 433^{2} + 23\cdot 433^{3} + 408\cdot 433^{4} +O\left(433^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 66 + 308\cdot 433 + 192\cdot 433^{2} + 273\cdot 433^{3} + 122\cdot 433^{4} +O\left(433^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 139 + 428\cdot 433 + 401\cdot 433^{2} + 176\cdot 433^{3} + 174\cdot 433^{4} +O\left(433^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 143 + 36\cdot 433 + 80\cdot 433^{2} + 92\cdot 433^{3} + 257\cdot 433^{4} +O\left(433^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $10$ |
$2$ |
$(1,2)$ |
$-2$ |
| $15$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
| $20$ |
$3$ |
$(1,2,3)$ |
$1$ |
| $30$ |
$4$ |
$(1,2,3,4)$ |
$0$ |
| $24$ |
$5$ |
$(1,2,3,4,5)$ |
$-1$ |
| $20$ |
$6$ |
$(1,2,3)(4,5)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
