Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 641 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 9 + 70\cdot 641 + 397\cdot 641^{2} + 444\cdot 641^{3} + 219\cdot 641^{4} +O\left(641^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 87 + 99\cdot 641 + 278\cdot 641^{2} + 628\cdot 641^{3} + 399\cdot 641^{4} +O\left(641^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 333 + 386\cdot 641 + 604\cdot 641^{2} + 407\cdot 641^{3} + 404\cdot 641^{4} +O\left(641^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 434 + 440\cdot 641 + 566\cdot 641^{2} + 485\cdot 641^{3} + 509\cdot 641^{4} +O\left(641^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 495 + 309\cdot 641 + 413\cdot 641^{2} + 522\cdot 641^{3} + 413\cdot 641^{4} +O\left(641^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 565 + 616\cdot 641 + 303\cdot 641^{2} + 74\cdot 641^{3} + 616\cdot 641^{4} +O\left(641^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,3)$ |
| $(1,2)(3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $5$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $2$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $72$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $72$ | $5$ | $(1,3,4,5,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
