Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 577 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 135 + 47\cdot 577 + 291\cdot 577^{2} + 559\cdot 577^{3} + 163\cdot 577^{4} +O\left(577^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 197 + 340\cdot 577 + 441\cdot 577^{2} + 384\cdot 577^{3} + 239\cdot 577^{4} +O\left(577^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 202 + 5\cdot 577 + 94\cdot 577^{2} + 575\cdot 577^{3} + 150\cdot 577^{4} +O\left(577^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 368 + 55\cdot 577 + 510\cdot 577^{2} + 404\cdot 577^{3} +O\left(577^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 382 + 576\cdot 577 + 564\cdot 577^{2} + 65\cdot 577^{3} + 262\cdot 577^{4} +O\left(577^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 448 + 128\cdot 577 + 406\cdot 577^{2} + 317\cdot 577^{3} + 336\cdot 577^{4} +O\left(577^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$5$ |
| $15$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$3$ |
| $15$ |
$2$ |
$(1,2)$ |
$-1$ |
| $45$ |
$2$ |
$(1,2)(3,4)$ |
$1$ |
| $40$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$2$ |
| $40$ |
$3$ |
$(1,2,3)$ |
$-1$ |
| $90$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$-1$ |
| $90$ |
$4$ |
$(1,2,3,4)$ |
$1$ |
| $144$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $120$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $120$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
