Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 571 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 38 + 184\cdot 571 + 300\cdot 571^{2} + 297\cdot 571^{3} + 112\cdot 571^{4} +O\left(571^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 358 + 364\cdot 571 + 173\cdot 571^{2} + 340\cdot 571^{3} + 203\cdot 571^{4} +O\left(571^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 399 + 394\cdot 571 + 557\cdot 571^{2} + 363\cdot 571^{3} + 514\cdot 571^{4} +O\left(571^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 412 + 212\cdot 571 + 558\cdot 571^{2} + 475\cdot 571^{3} + 185\cdot 571^{4} +O\left(571^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 508 + 556\cdot 571 + 122\cdot 571^{2} + 235\cdot 571^{3} + 125\cdot 571^{4} +O\left(571^{ 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 value |
| $1$ | $1$ | $()$ | $4$ |
| $10$ | $2$ | $(1,2)$ | $-2$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $20$ | $3$ | $(1,2,3)$ | $1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $0$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
