Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 593 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 57 + 169\cdot 593 + 591\cdot 593^{2} + 360\cdot 593^{3} + 494\cdot 593^{4} +O\left(593^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 108 + 582\cdot 593 + 186\cdot 593^{2} + 531\cdot 593^{3} + 476\cdot 593^{4} +O\left(593^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 234 + 503\cdot 593 + 12\cdot 593^{2} + 45\cdot 593^{3} + 163\cdot 593^{4} +O\left(593^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 271 + 434\cdot 593 + 120\cdot 593^{2} + 425\cdot 593^{3} + 251\cdot 593^{4} +O\left(593^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 308 + 425\cdot 593 + 524\cdot 593^{2} + 547\cdot 593^{3} + 73\cdot 593^{4} +O\left(593^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 380 + 43\cdot 593 + 250\cdot 593^{2} + 221\cdot 593^{3} + 528\cdot 593^{4} +O\left(593^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 479 + 538\cdot 593 + 223\cdot 593^{2} + 178\cdot 593^{3} + 504\cdot 593^{4} +O\left(593^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 537 + 267\cdot 593 + 461\cdot 593^{2} + 61\cdot 593^{3} + 472\cdot 593^{4} +O\left(593^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,6)(2,8)(3,5)(4,7)$ |
| $(1,4)(2,5)(3,8)(6,7)$ |
| $(1,6,4,7)(3,8)$ |
| $(1,5,4,2)(3,7,8,6)$ |
| $(1,4)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,4)(2,5)(3,8)(6,7)$ |
$-4$ |
| $2$ |
$2$ |
$(1,6)(2,8)(3,5)(4,7)$ |
$0$ |
| $2$ |
$2$ |
$(1,7)(2,8)(3,5)(4,6)$ |
$0$ |
| $2$ |
$2$ |
$(1,4)(6,7)$ |
$0$ |
| $4$ |
$2$ |
$(1,8)(2,6)(3,4)(5,7)$ |
$0$ |
| $4$ |
$4$ |
$(1,5,4,2)(3,7,8,6)$ |
$0$ |
| $4$ |
$4$ |
$(1,5,7,3)(2,6,8,4)$ |
$0$ |
| $4$ |
$4$ |
$(1,3,7,5)(2,4,8,6)$ |
$0$ |
| $4$ |
$4$ |
$(1,6,4,7)(3,8)$ |
$0$ |
| $4$ |
$4$ |
$(1,7,4,6)(3,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
